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With Rhaenyra dead from such an early age from childbirth, Visery's health would probably overall decline much quicker through grief, with that said he is still relatively active and present at court at the time Luke is born and therefore in a better position to try and support Jace/Luke's claims if he still wishes to name Rhaenyra's heirs as his own. Its possible he makes the lords swear to them as he once made them do so for their mother. Jace as his "heir" in this scenario will be made Prince of Dragonstone and will either be given command of the island, or he may wish to keep his daughter's son close to himself in his grief and out of a desire of teaching him to rule in the absence of his mother. Whether the Hightower's encourage this or speaks to their aims, if they raise minimum debate about this then it means they intend to keep Jace close to assassinate him/capture him when Viserys dies. If they protest it strongly they fear him becoming popular in court. Jace is a natural commander, likeable, charismatic - he would earn himself allies in court and his being a man would put some people at ease. Likewise Leanor would still be alive and wouldn't abandon his children/be assassinated by Daemon so he would likely stay with Jace in either Kings Landing or Dragonstone to support his son. Though it is also possible that Corlys stays with Jace to support him/his claim and Laenor stays in Driftmark with Luke, as Laenor will be Lord of the Tides and Luke will follow him in this role, while Corlys is the big political thinker and would be of more use to Jace. If he is in KL Corlys/Leanor providing support will strengthen Jace's claims. Luke is almost certainly raised in Driftmark by Leanor/Corlys to act as his heir, he would make allies of his own as Corlys consistently supports Luke as his heir and grooms him for the role - Luke is likely a competent sailor, commander and lord by a young age and is well-established as Corlys' heir after Leanor.
This brings us to the biggest problems for the Velayron boys - that they appear to be bastards. The Greens will use this against them at every turn and try to implore to the realm that a bastard cannot sit the Iron Throne, and they would likely try to convince even the lords who swore and remain loyal to Rhaenyra that her legitimate line is ended and that Aegon would be her "heir". Many lords, especially those of the Green faction, are persuaded by this argument and place their support behind Aegon. Viserys almost certainly has any suggestion of Jace/Luke's bastardry made punishable by death due to it besmirching his beloved dead daughter's name. Likewise many of Rhaenyra's loyalists would likely be incensed by this defamation.
The Hightowers would try to convince Viserys that he needs to put one of his own children on the throne, "for the stability of the realm!" and have him declare for Aegon (I personally see this as unlikely). They may try to use personal manipulation on him by pointing out how young Jace/Luke are and suggest that the throne may crush them. They would likely try to persuade him to allow Jace Dragonstone and Luke Driftmark and give Aegon the Iron Throne. Again I see this as unlikely but he may genuinely consider it for a time, incorrectly believing it would be the path that best protected his family from the vipers in the nest.
The reason this plot would fail, and the reason Jace would have a surprising amount of influence at court is that Corlys would be 100% dedicated to putting his grandson on the throne and would likely push for Laenor/Rhaenys/himself to have a seat on the Small Council to protect their interests, as afterall, the first "son" of House Velayron is now the heir to the Iron Thone. Removing Jace/Luke from the line of succession would be tantamount to declaring war on Driftmark, it would be an unforgiveable insult to Corlys and would "legitimize" the "vile rumours" made about the beloved late Princess Rhaenyra by her enemies, it would also effectively rob Corlys of his own heirs/line.
Unlike Rhaenyra I doubt Corlys would simply ignore the situation in KL and would likely actively try to undermine the Hightower influence by any means in order to protect his grandson's legacy. On that note, Corlys, Rhaenys would likely be far closer to and more invested in Jace/Luke in this timeline as they would have raised the boys themselves alongside Laenor.
Daemon would have less of a reason to stay in Westeros, assuming Laena dies the same way and may return to Pentos, however it is likely that Corlys will persuade Daemon to his side by both appealing to his hatred for the Greens, his bitter regrets over Rhaenyra, and most importantly by suggesting the marriage pact between Jace/Luke and Daemon's daughters. Effectively offering to make Rhaena Lady of Driftmark and Baela Queen of the Seven Kingdoms in exchange for his supporting Corlys in making Jace king. He would likely accept.
Daemon would likely return to court as Corly's urging and to Viserys relief - he loves his brother and would gladly welcome him with open arms, its one of the few things I doubt Otto or Alicent would be able to persuade Viserys on. Viserys likely gives Daemon a high position in court, and eventually a seat on the Small Council, he consistently weakens Otto's grip on the council alongside Corlys (or whomever Corlys has in his place - probably Laenor or Rhaenys) and Daemon would increase his own influence in the city and weaken the Greens and their allies. Otto would fixate on removing Daemon, but Viserys - increasingly weak and ill would rather remain on good terms with the most dearest loved one he has left and simply let Otto and Daemon vent their "frustrations" at each other. He may also seek to discreetly arrange "accidents" for minor Green supporters like Cole, against whom he would have a grudge. He would be unlikely to assassinate Otto or other big Green figures while Viserys lives.
Would the eye incident still happen... hard to say, but probably. Laena's death probably still happens the same way, her funeral is still held on Driftmark, Vhagar is still there and Aemond is still without a dragon. Rhaena is probably still wroth that Aemond claimed her mother's dragon and the other children likely still support her. The tension around Jace/Luke being bastards is a lot stronger in this timeline as it is the main weapon the Green's have to delegitimize their competition, so Jace is likely still driven to anger enough to pull a knife and from there I will assume all takes the same shape of events. Of course lots of things could change, afterall Luke would be likely living in Driftmark as its future heir and would therefore have his own personal guard looking afterhim rather than the seemingly negligent Kingsguard so maybe he isn't present? Likewise Corlys' guards might be more diligent with the heir to the throne than the Kingsguard? Maybe Daemon is keeping a closer eye on his daughter's without conspiring with Rhaenyra to wed and be rid of Laenor. Maybe Harwin is still alive and keeps a close enough eye on his late charge's sons (his sons) that they don't get into such mischief. Who knows?
As for Harwin, my personal belief is that he still dies - Larys had more to gain from murdering his father and brother than just pleasing Alicent, Harrenhall for instance... So in all liklihood he is still killed alongside Lord Lyonel. Though its hard to say if they die in the same way/time as Alicent's request likely sped up Larys timeline and Lyonel resigning as Hand was due to pressure from accusations of bastardry that no doubt still exist, but might not be quite as open considering that Viserys would react far more harshly and there is no Rhaenyra at Court to stir them up. But then perhaps the rumours are even worse? Like I said earlier the bastardry claims are now the chief weapon of the Greens in disinheriting Jace. Depending on timing its possible Otto isn't restored as Hand quite as quickly as he was originally, in which case he has less time to take full control of the Small Council before Visery's death - especially with Velayron influence over the years increasing and attempting to strangle out any Green influence.
The relationship between Alicent and Rhaenyra's sons is.... complicated. On one hand it is MUCH harder to paint little boys as the ruthless arch-villain Otto painted Rhaenyra as, and saying a young boy would have you and your children murdered to secure his claim wouldn't be too convincing - he would likely play up Corlys' involvement and ruthlessness and once Daemon started supporting Corlys it would be easy to suggest Daemon would simply kill them all to secure his daughter's and her betrothed's thrones.... and for once he may even be right. On the other hand these boys are all thats left of her once cherished friend, and they are, ultimately, children. Plotting against them would feel very different from plotting against Rhaenyra and her faction. On the other hand, they are bastards "stealing" her own children's birthright by any law of the land, as even IF you still saw Rhaenyra's claims as legitimate her children are NOT, and thereby Rhaenyra's legal heir would be Aegon, so the indignity would likely drive her forward, though she would almost certainly not allow Rhaenyra's sons to be harmed, especially if war breaks out earlier when they are still undeniably children in the eyes of the realm.
How Alicent sees them drastically affects how her children see Rhaenyra's - in all likelihood they too have a complicated relationship - more distant from Luke who is likely raised in Driftmark, and possibly love/hate/complicated with Jace who is raised at KL with them for much longer than in the original timeline, or again very distant with Jace being raised in Dragonstone/Driftmark. The fact that their nephews are bastards would be drilled into them at every opportunity by Otto and he would likely reinforce that Jace is stealing their birthright. If the eye incident still happens then Aemond hates whichever Velayron boy/girl inflicted the wound and hates Jace/The Blacks more generally.
Viserys would die earlier in this timeline, the grief of losing his favourite daughter would quicken his decline.
Make no mistake - war still ensues..
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Depending on how MUCH earlier he dies its possible that many of the main characters of the conflict are still children or early teenagers. If it is before the eye incident/Laena's death, then clearly that no longer happens and the Greens lack any adult dragon riders and only have Sunfyre and Dreamfyre at best to match Laenor's Seasmoke and Rhaenys' Meleys (if Laena and Daemon remain neutral) and if they join then Caraxes and Vhegar are fighting for the Blacks also. This is a suicidal prospect and the Greens would be utterly outmatched in dragons/riders (especially riders) and would likely try to win the Throne through a quick coup/assassination attempt or warfare/politics.
Warfare is useless, as the Black's dragons would simply make a Field of Fire out of them. Likewise in terms of navy the Velayron's have any of the Green's allies beaten easily. Likewise, against this type of firepower its very, very unlikely that other lords of the realm would be keen to join the side of the conflict with zero (maybe 1 or 2 depending on Aegon and Helena's age/ability) dragonriders and the weaker navy.
Its likely then that Otto would suppress news of Visery's death and try to quickly capture assassinate Jace, Corlys, Laenor, Rhaenys, Daemon (he wouldn't even try to capture Daemon, he is too dangerous to live and would never submit) if they remain in KL and if they are not he would send assassins to Driftmark to try and kill them. Alicent would no doubt be dead against this but given the sheer disparity in firepower its unlikely Otto listens to her - he knows the Velaryons will never accept terms when they hold the clear and undeniable advantage through dragons.
However I doubt the success of this attempt. If it takes place in KL he has better odds, but Daemon/Corlys are no fools and Daemon in particular has allies amongst the cutthroats and City Guard - they would only be able to take him by surprise by using the Kingsguard and it is likely that given such an order by Otto that many of the White cloaks may defect and warn them rather than take place in such a dishonorable and evil act. Daemon would likely have more support in the City than Otto and if Corlys brought his own men/lent them to the defence of his son/grandson in the Capital than it is likely the Red Keep becomes a hive of fighting as every faction fight each other and their own ranks - divided in their loyalties...
If The Blacks win this skirmish then the war is won, Otto and conspirators are dead and then there would be a debate what to do with Alicent/her children (given that most of them are genuinely children at this point it would be difficult for even Daemon's reputation to survive murdering them all - though I have no doubt he would argue for that, Leanor/Jace would probably advocate for mercy, Corlys is hard to decide but probably mercy also so long as his son/wife/grandchildren are unharmed). Aegon might be executed as a usurper, though he would actually be more valuable alive and openly renouncing his claims. He, Aemond and Dearon would perhaps be kept as hostages or would be given to the Wall/Citadel to prevent their claims from being supported. Helena is either kept as a valuable bargaining chip or made a Septa. Given Jace is merely 6 at this age then Corlys/Leanor/Daemon likely set up a Regency Council and rule in his stead/raise him for the role until Jace comes of age. There is no real war, the transition is pretty painless and the realm moves on relatively quickly though Green aligned Lords would occassionally whisper for the restoration of their chosen heir. But their time has mostly passed and The Regency Council would be ruthless in rooting out all treason and conspirators given the violent attempt at a coup the Hightowers attempted (House Hightower itself likely faces harsh penalities and may even be stripped of its possessions given the undeniable proximity to the plot through Otto & Alicent - the Lannisters are likewise punished for their role in supporting the Usurper though I doubt they would be outright dispossessed of Casterly Rock/The Westerlands).
If the Greens win this skirmish then this is only the beginning. Depending on who is at the Capital its possible Jace/Leanor/Daemon/Corlys/Rhaenys are dead - there would only be need for one of the Velayron's to be present with Jace + Daemon so two out of either Corlys, Leanor or Rhaenys are likely alive and in Driftmark with Luke. Otto would no doubt try to send assassins for the rest but this is likely unsuccessful as Driftmark would be on high alert once contact from KL stops, if their own spies don't outright inform them of the treachery. Corlys/Leanor/Rhaenys declares Lucerys King of the Seven Kingdoms and instantly calls their banners to war and prepare to besiege KL, they inform the Realm about the Crimes of the Hightower Plot and declare Aegon a usurper and Kinslayer and ask all noble lords declare themselves, including helpful reminders that despite the Hightowers's murder of Jace/Daemon/X that they still possess dragons with adult riders and the Greens do not. Arryn and Stark would certainly side with them instantly, followed by lords who are both appalled by the murders in KL and those who know to side with the faction with dragons. they do not hesistate to strike hard at the Capital with dragonfire and their superior navy, the city likely falls in a two days or less - any and all involved in the Hightower plot are killed, including Aegon and possibly Aemond, Helena/Dearon is likely kept as hostages though if they feel ruthless they may kill Dearon also to prevent any male claimants from Alicent's line, strengthening Lucery's position. If the Greens are wise and flee the Capital then the Velayron army/navy/dragons simply chase them down to wherever they flee/make their stand (likely Oldtown or Lannisport/Casterly Rock) and destroy them there after some bitter fighting. The War is brutal, bloody but ultimately swift - and the kingdom remains relatively unscathed (with KL taking the brunt of it), Lucerys is king but Luke is too young to be King regnant at this point in time (Afterall the war has happened in this version when he is but 5 years old before he cut Aemond's eye) so Leanor or Corlys is made his regent and molds him for the role. As soon as Lucerys comes of age there is significant pressure on him and his wife (likely still Rhaena but there may be some push to make it Helena thus uniting the last potential remaining claims to the throne - and conveniently increasing the chance of the kids having the all-important Targaryean look) to produce heirs quickly to secure his line and produce a new heir for Driftmark. The Hightowers are absolutely removed from their seat of power/probably executed and replaced with an allied House of the Blacks/Velayrons. Other Houses who sided with the Usurper are likely harshly punished if not outright executed - Lannister would have a reckoning, though due to their power they may get away with simply allowing Tyland to be executed and pin their involvement on him. Its a bloody few years by Luke's regent to assure his throne and by the time Luke comes of age he is likely unopposed.
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If Viserys manages to hold on for a few more years until Aemond has/is able to control Vhagar, Sunfyre/Aegon are mature dragonriders (maybe even Dearon and Tessarion) then there are some key changes that might drastically alter things....
But with that said - its not even a guarantee in this scenario that Aemond claims Vhagar at all, given that with Rhaenyra dead from birthing her 2nd child, Syrax is now riderless and resides in King's Landing. Aemond would have far more opportunities to try and bond with this dragon and I have no doubt he would try. There is no reason to favour him failing or succeeding in claiming Syrax as we know some dragons (like Vhagar) are able to move onto a new rider relatively quickly after the death of their rider (who died in the same circumstances no less) and others take more time. If Aemond fails to bond with Syrax then things progress as they did originally.
If Aemond claims Syrax then this is a game changer. Aemond would have claimed his dragon younger than he did in the Original timeline, this would drastically alter his personality as he is now no longer as hungry to prove himself, is more assured, and wouldn't face the same bullying for being dragonless that he did originally. His relationship with Jace is likely a bit strained due to him claiming his dead mother's dragon, but it is extremely unlikely that Aemond loses an eye in claiming his dragon as the circumstances of Driftmark were so specific that I doubt they would be repeated verbatim. This again changes Aemond, who no longer holds such deep hatred for the Velayron boys in his heart as he has not been maimed by them, they have "taken" nothing from him - as the second son he never stood to inherit the throne the bastards are "stealing" and they have never personally wronged him, whatever their relationship may look like. This makes it FAR less likely for Aemond to lose control and be the one to "ignite" the war.
The biggest difference here is that Aemond no longer controls Vhagar, this deprives the Greens of their greatest weapon and trump card. Syrax may be an adult dragon, but she is NO Vhagar and would be easily outclassed by Meleys, Caraxes or Seasmoke - all greater dragons that are actually used to battle. This greatly changes the balance of power with the Dragonriders in the war - the Blacks are the clear superiors against Dreamfyre (who while older is small for her age and notably less fierce than most dragons), Sunfyre (fierce and large for his age, but no match for Meleys or Caraxes - might win against Seasmoke hard to tell, especially with Leanor as his rider) and Tessarion (again no match for Meleys/Caraxes, drew in the fight against Seasmoke in canon, probably loses now due to Laenor's superior experience as a rider). With this in mind, The Blacks have every reason to bring the fight on dragonback to the Greens. In all likelihood the Blacks will win outright. It is also likely more lords would back the Blacks' as without the Green's possessing Vhagar its clear the balance of power with the dragons favors the Black faction.
The Dragonseeds are likely never needed or used, its Jace's plan and as the war breaks out earlier in this time he is maybe too young to think of it. If Aemond has Syrax in this timeline then the Dragonseeds are not needed by the Blacks at all to counter Vhagar and they would constitute an unnecessary risk.
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If Vizzy T lives long enough for Aemond/Dearon to become proficient dragonriders then things get messy.
On one hand if Daemon/Corlys/Jace are at court all these years then things would be resolved by a quick burst of violence in the capital like I describe above and would follow said scenarios for the most part. With both sides trying to prevent the other's dragonriders reaching the Dragonpit and turning the tide of the fight.
If Jace/Corlys/Daemon are at Dragonstone/Driftmark then it is tempting to say this goes similar to canon, but in truth there would be some major differences.
Rhaenyra was the one who delayed action at the beginning of the Dance, with her gone Daemon and Corlys would be ruthless in pushing their claim - its possible he pulls off a Blood and Cheese moment with the aim of killing Aegon's entire line. Its hard to predict who/what would happen in such a attack or even if Daemon would do it - as it requires a deeper knowledge of his internal compass than we have. Does he kill Jaeherys only out of sadism or because its an equal exchange of sons... would he take the step to become a kinslayer without such a provocation like Lucerys' death? hard to say. Either way its unlikely he manages to have the entirety of the Greens assassinated if he decides to go this route. Depending on this attack/its success then the Greens may respond by attacking either Dragonstone or Driftmark directly with their dragons out of revenge, resulting in a battle in the skies over the islands. Assuming all the Black Dragons are either at Dragonstone or Driftmark the balance of the fight is on their side, they may repel Vhagar and Sunfyre (Dreamfyre is unlikely to take part as Helena would be in no state and Dearon/Tessarion are too far away for a quick strike) but the devastation is great and possibly more than one dragonrider would die on both sides. Either way its likely that Aemond/Vhagar survive unless more than half the Black Dragons are in one location, in which case even Vhagar is easily outmatched. If the Black dragons are split up between Dragonstone/Driftmark or elsewhere and Vhagar/Sunfyre attack in unison than the island in question is likely heavily damaged and the Blacks suffer a blow. Though Meleys/Arrax and Caraxes/Vermax with Seasmoke potentially being in either location still puts the balance of the fight with the Blacks. Its possible Otto/Alicent are able to persuade them not to attack due to the odds being against them despite their losses/grief.
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The Dragon messengers are now unlikely to occur - Jace as the future king is too valuable to risk, Lucerys may still serve as a messenger but its unlikely as the only other claimant the Blacks possess. Both can only be risked when no other options exist. For this reason the fight above Shipbreaker Bay is very unlikely to happen. Especially as Deamon/Corlys are unlikely to simply believe reminding lords of their oaths to be enough to change the balance of power - they likely send ravens, with offers/terms, to houses but make no mistake they would sooner act first then be caught unawares. For this reason Blood and Cheese will not occur as "revenge" for Lucerys - but as I mentioned above this doesnt mean Daemon doesnt use them more proactively.
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In any case Daemon takes Harrenhall and more quickly. The Battle of Burning Mill/Taking of Stone Hedge is still a victory for the Blacks. Giving the Blacks control over the Riverlands early on in the war. Likewise the Sack of Duskendale likely is a victory for the Green's as it was in the original timeline.
The Battle of Rooks Rest is likely very different. Corlys in the original timeline blamed Rhaenyra for Rhaenys/Meleys death as he argued that a force of dragons should be deployed. Given he is effectively head of the Blacks in this timeline he likely sends both Rhaenys/Meleys + Laenor/Seasmoke + Luke/Arrax or Jace/Vermax (makes no sense to send both and endanger both heirs) Maybe even Daemon/Caraxes given HarrenHall isn't that much further away then Dragonstone/Driftmark. Given that Rhaenys/Meleys on their own were nearly a match for both Vhagar & Sunfyre and left Aegon/Sunfyre crippled for life, I feel confident in saying the Blacks win this battle pretty easily - Aegon/Sunfyre likely are outright killed and there is a good chance Aemond/Vhagar also fall or are heavilly injured and forced to withdraw. Cole and his forces are likely destroyed outright. This is a major defeat for the Greens and all but guarantees their loss. If Aegon and Cole are dead and Aemond escapes (injured or not) then Jaehaerys is named King and Aemond becomes his Hand, regent and Lord Protector. If Aemond is dead then Otto is reinstated as Hand and Alicent is made Regent and Daeron is Lord Protector.
Needless to say the Battle of the Gullet goes very, very differently - in that it likely doesn't happen at all. No Aegon the Younger and No Viserys means there is no need to secure them in Pentos, they are not attacked by the Triarchy and Jace doesn't go to save them. Likewise Jace won't likely be allowed to fight directly so he isn't killed by crossbowmen - anyother dragon rider is unlikely to die as Jace is the only dragonrider in the fight killed due to flying too low - a mistake the others avoid bar Stormcloud (who died escaping with Aegon) who won't even be there due to his rider Aegon the Younger not existing....
With Aegon/Sunyre and potentially Aemond/Vhagar neutralized then the Blacks will likely march on KL much sooner and take the city.
If the big dragon battle happens so early in the its very likely that the Sowing of the Dragonseeds is avoided as the death Lucerys/Arrax and then the death of Rhaenys/Meleys at Rook's Rest inspired the move. Even if it still happens they would likely be more loyal to Jace, who raised them up, than to Rhaenyra and if the 2 main Green Dragons are already dead or disabled then they have less reason to defect. But overall the Sowing almost certainly doesn't happen if Rook's Rest is a Black Victory.
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If Aemond survives and is injured alongside Vhagar then he likely pushes an aggressive strategy but his ability to do so is now greatly diminished with his dragon potentially out of commission and their forces destroyed at Rook's Rest. If he and Vhagar are dead then the ability of the Greens to be proactive has all but been demolished, especially if Aegon/Sunfyre is also dead/injured. Daeron and Tessarion might be called to KL to protect the city from Dragon attack alongside Dreamfyre. Hard to say as there are several points in the canon timeline Daeron could have been used but wasn't due to him being seen as too young by his older relatives, but in such a desparate scenario that may no longer matter. If Otto and Alicent are the ones now in charge then Otto pushes for more allies, making desparate offers to any and all who will listen in the hopes of overwhelming the Blacks despite their superior number of dragons. If The Blacks lose key dragons at Rook's Rest then the Greens will still have some hope and would likely double down on attempts to assassinate the Black riders and call in outside help like the Triarchy. They would certainly push for Baratheons and the Lannisters to be more proactive in supporting the Capital.
If Aegon/Sunfyre + Aemond/Vhagar did the smart thing and fled then the war for the Green's isn't lost - but their military might, commander and Hand (Cole) are now dead. That Rook's Rest was a disaster will be impossible to deny and Lords wavering in the middle will see the slew of Black victories in succession and determine them to be the winning side. Far more Houses will declare for the Velayron claimant and the momentum of war is on the Black's side. It is possible that the Blacks, now in firm control over the Riverlands and with a decent foothold in the Crownlands, take the fight to KL and the result is a mighty and bloody battle. More likely they will attack Green allies in the Crownlands and cut off the city via land and sea and use their dragons to protect their troops - starving the city and forcing the Greens to either flee, attack their positions (putting them at a disadvantage) or negotiate a surrender (which the main Green figureheads will be against as the chances of them being spared are little). With the Riverlands secured and victory increasingly on the Blacks side the armies of their allies may move a little quicker to support in order to end the war as quickly as possible. The North and Vale begin marching to support Daemon's Riverlands forces and march on the Capital. Though they may clash with Green forces on the way - with Daemon supporting them they would win rather handily. Or, they take the same amount of time as they do in canon and the war enters a quiet period while the Greens recover and try to decide their next move and the Blacks summon their forces for a final push. This likely ends with the City starving weakening the hold the Greens have on the city and the Greens putting up one last battle to break through and flee West to the Reach/Westerlands that may or may not succeed depending on which of their dragons yet live.
If they make it then the Greens call their allies to them, they regroup and prepare to fight to the last - the Blacks take the City install Jace officially as king, remove their blockade and food begins reaching the city again, quelling tensions for a time. Daemon either remains in the city with Jace to enforce order or he leads the pursuit of the Greens (seems more likely to me) and Corlys takes control over the city and secures it.
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Its unlikely that the Riot of Kings Landing and the Sacking of the Dragonpit still occurs - it was precipitated by Rhaenyra's paranoia - she turned on Corlys after he warned Addam Velaryon that Rhaenyra intended to imprison/execute them and the other Dragonseeds following the betrayal of Hugh and Ulf at Tumbleton, resulting in the Velayron forces (over 1/2 her army) abandoning her. Well considering there is no Rhaenyra, possibly no dragonseeds, and Corlys certainly isn't going to betray himself... Plus given Helena would have fled with the Greens then she isn't in the city to commit suicide and spark the riot by rumours of her murder (which was believed due to the monstrous murder of her child by Blood and Cheese). SO even IF the Shepard still stirs up the people (potentially with Larys' aid) the sparking event is gone and the city has the forces to enforce order.
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With the City taken in this little hypothetical scenario then the war is winding to a close. If the Greens didn't flee successfully then Corlys/Daemon/Laenor/Jace have decisions to make. Aegon, Aemond, Otto, Alicent, Larys, Tyland Lannister, Jasper Wylde and Orwyle are likely executed nearly immediately (Tyland might be held hostage to force the Lannisters to ceasefire and withdraw - Jason might do so if his twin's life is assured - if this happens then Tyland will eventually be sent to the Wall) if they didn't die in the city's fall. If Daeron had been summoned to the city he may also be killed - depending on if he surrenders/swears fealty, how active in the fighting he is, if he bends the knee or is seen as too young to be executed then he is sent to the wall, most likely though he would have died in the fighting given he is a dragonrider (possibly the ONLY combat-capable Green dragonrider at this point) facing far more experienced dragons and riders. Aegon's children and Helena are held hostage till the war's end (again assuming they all live, Helena may be forced to fight on Dreamfyre).
If the Greens escaped then they meet up in the Reach with Dearon/Tessarion (if he isn't part of their escape) and the main body of their forces gathered by the Hightowers, Lannisters and Baratheons. The area is more firmly in their control and they likely call all their forces allies forth with the intent of drawing the Black's into battle - they will try to fight on many fronts so that the Blacks dragons do not simply annihilate their forces (Afterall they can't be everywhere at once), What follows is a period of protracted fighting in the Reach and Westerlands and other Green held lands. If the Blacks are smart they will devastate the Stormlands and Storms end with Dragonfire - forcing the Greens to either move one of their few dragons to the area to defend it, or more likely, forcing Borros to withdraw his forces to his own lands. Repeating this move with the Westerlands may prove equally effective - though the Lannisters are more invested in Green victory at this point than Baratheon. If he still lives Daemon would probably choose to cut this protacted fighting short by taking Caraxes and his forces against OldTown - the seat of the Green's power, forcing them to regroup to defend it.
Depending on which Dragons live at this point this becomes the last great battle of the war, if the Greens still have Vhagar, Sunfyre, Tessarion and Dreamfyre this could match Caraxes, Meleys, Seasmoke, Vermax (again I dont consider this likely due to threat to the heir) or Arrax, (Baela's dragon Moondancer is probably still too young for battle). If the Greens have lost Vhagar then the battle favours the Blacks, even if the Blacks have lost Meleys and Caraxes at this point. If Caraxes, Meleys, Sunfyre and Vhagar are gone then the fight still favours the Blacks as Seasmoke is a veteran dragon/rider combo against Tessarion/Daemon + Helena/Dreamfyre with either Arrax or Vermax as support. If Vhagar is the only living Green Dragon she will win against the Blacks if both Meleys and Caraxes are dead. If either lives then frankly Vhagar probably loses but its a bloodbath and a pyric victory for the Blacks.
If the Blacks win then the war is over, their victory complete, House Hightower is likely extinguished for its crimes and the Houses Lannister, Baratheon and other notable Green Houses are punished severely if not outright dispossessed. The odds, considering the chain of events getting us here, favour a Black victory (As we see in canon) but the war has been devastating for the country and House Targaryen.
If the Greens win this fight (somehow), then the war is still ongoing, most likely both sides dragons have been exhausted, are either dead or injured beyond recovery. And the war leaves the nation cleaved in two with the North, Vale, Iron Islands, Crownlands, Riverlands being under Black control rule and the Reach, Westerlands, Stormlands supporting the Greens. Jaehaerys is proclaimed the True King in exile from OldTown and the Blacks declare Jace the True King on the Iron Throne - both are too young to rule outright, but Jace is significantly older than the infant rival king and rules from the seat of power in the realm, many lords support him as the true ruler and his claim appears stronger, especially as Borros (fairweather friend he is) increasingly sees he backed the wrong horse and tries to distance himself from the Greens. Lords on both sides will want the war to come to an end. The Blacks probably have the Greyjoys sack Lannisport/attack the Lannister and Redwyne fleets with the Velayron fleets support - annihilating the Green's naval power and attacking the seat of Lannister power and blockading Oldtown itself in time. The Blacks, now lacking dragons bar those of the King Jace and Queen Baela, either attack directly (if the Green dragons are all incapacitated) or resort to traditonal warfare to win outright, attacks on the nation's breadbasket as Winter approaches means the Blacks must import more food, raising the realm's debt - the situation cannot continue so the Blacks will probably gather their main force, now bolstered by the full Northern and Reach armies to either take OldTown once and for all or force the remaining Greens to at last accept defeat.
Either way there is no way out for the Greens, they either die fighting or surrender - Aegon's children are spared and are either sent to the Wall or are kept hostage at court and are eventually married off to Jace's own children. Green Lords are either killed or given VERY unfavorable terms depending on the strength of the Black's position and when/if the Greens finally surrendered.
Ending the succession crisis and the Dance of the Dragons once and for all.