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What if Catelyn Stark arrives at the inn one day after Tyrion Lannister?
Well the first impact that this has on the story line is that Jaime has no reason to go after Ned Stark. Ned wasn't the great fighter he is portrayed in the show, but he is no slouch either. While Jaime could defeat Ned, the average City Watchman would not be able to. If the throne room scene goes down the same way, and Ned fights his way free we suddenly have a very different story. Catelyn and Ned reunite and make another baby Stark.
Ned fights his way outside the city, and makes his way back North to Winterfell. With Stannis and Renly both declaring themselves King, I see no reason why any of Joffrey's advisers would be in favor of antagonizing the north further. Sansa and Arya are returned to the North, and the Starks essentially secede from the rest of Westeros.
This scenario is rather like Dorne, and the Starks declare themselves neutral in the war of 3 Kings. The Greyjoy Rebellion part 2 never occurs and Theon Greyjoy likely marries Sansa Stark to keep the peace between the North and the Iron Islands. Now their heirs have a claim to Pyke and Winterfell.
Tyrion's fate isn't as good in this scenario. Without Bronn and his clansmen, he likely is killed when King's Landing eventually falls to the combined Baratheon/Tyrell army. I feel that without Catelyn Stark's presence at the meeting between brothers, they begin their fight earlier than in the original time line. Renly attacks Stannis or Stannis attacks Renly without the shadow demon and Stannis falls. Once Stannis falls, Renly feels that the throne will be his, and they move quickly. Renly's massive host descends on King's Landing and sacks the city. Even Tywin Lannister is not going to be able to defeat an army of that size.
Tywin has a further problem, without mobilizing his forces against the Starks, his army isn't nearly as prepared for a war as it would be with the direct threat of Robb Stark's army. Even a master strategist like Tywin isn't going to have the ability to muster his full army quickly. So when he does march on King's Landing, he is at less than full strength. His hand is forced by the timetable that has been set by Renly Baratheon.
In Summary, Renly Baratheon takes the Iron Throne, House Lannister becomes extinct in the main line House Stark drifts away from the rest of the Seven Kingdoms, House Tyrell has the only Queen in Westeros, House Greyjoy rules Pyke from Winterfell, and House Targaryen faces a nearly unified Westeros if Dany decided to invade. One day difference has changed the entire outlook of Westeros!