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HxH timeline: 2006, BSD timeline: 2012. (Or those reaction writers ignore it completely as if it were 2020+)
Background: Same as Killua has it. After Gon separated with his father, he and Alluka go on to Yokohama, seeking for purpose and self reliance. Killua 14, Alluka 10.
During Yokohama: The headmaster discovered him and Alluka, adopting them into the orphanage. Once they learned the unfair treatment of this place, Killua arranged a wish to reset the standards, Nanika of course doing it. Soon the headmaster has a stern yet caring persona, and the kids are saved from further damage.
Naming: Killua wanted to change his name, The headmaster suggests "Atsushi", then named Alluka “Kyouka”. Their surname from the headmaster, "Nakajima". (They kept their Zoldyck surname a secret)
Shibusawa appearance: A year in, he adopts Atsushi in his respective fortress to obtain gifted crystals, looking for the pinnacle of all abilities. Those victims lose their memories when the crystals are collected, and some lose their lives. All with his NEN, which Shibusawa didn’t know of until he explained about it, and he wasn't interested in possessing any luckily.
Even though he stays there, Atsushi still visited the orphanage now and then. His last name is kept "Nakajima."
Meanwhile for Kyouka; she is also adopted to the Izumi family the same year at 11 years old. Unfortunately her parents didn't last long the next 2 years. Later rejoined into the Port Mafia instead. Atsushi didn't know this. The deceased parents weren't news mentioned, cleaned up the home, and kept secretly due to government and Mafia reasons. The headmaster himself informed Atsushi only about Kyoka's family "moving" someplace else, he doesn't know either.
Kyouka ability explanation: As Alluka, her Nen comes from Nanika. It takes three requests to awaken Nanika, three or four to be evaporated if you disobeyed Alluka. But here, she has grown and changed. As Kyouka, it doesn't matter how many requests or disobedience they give her, Nanika occurs only if Alluka wanted her too. Ever since her adopted parents, she received demon snow in a cellphone by her mother, and Nanika stays too.
The Port Mafia didn't know of her until Akutagawa brought on her wrath, and a bonus to her growth, she can command a wish to Nanika telepathically. She didn't feel comfy with most of them to let anyone have a wish of their own. Her birthday is celebrated there and remained for 6 months.
Atsushi ability explanation: Little ol' Atsushi had weretiger powers since his association with Shibusawa. He was just cursed a while back without knowing it, and Shibusawa made the electrocution scene happen, dying as a result. But there wasn't only a scratch, it was a splattered mess! Because he got family and hunter trauma caught up to him. He was 17.
Aftermath: Hid the man within the fortress, and stayed back at the orphanage until 18 and left to find a job.
BSD first chapter/episode: He's on the bridge thinking about a career as an author, what'dya know! Then the rest of the story goes on; Saving dazai, being investigated at the shop, puts on an act for his tiger issue, Dazai already knows and in the end, he becomes a detective instead! Unwillingly... but grateful.
Appearance: Same uniform from the agency. Same build. Different hair and eyes. Haircut was done as a prank by the kids, he kept it. After the Port Mafia's tiger attempt, he actually goes to a hairdresser and evens his bangs. His eyes and hair colors changed due to the weretiger colors, or you can pretend his hair is mixed, and the eyes casually a blue and yellow.
Kyouka's appearance: Black hair color kept, no dark blue hue. Blue eyes no different. The hair accessories were removed at the Izumi household, so does the shrine maiden outfit. At the Mafia given her red kimono by Koyo, but she did join the ADA, so Fukuzawa gave her a different kimono. A light blue kimono with a pink haori, yellow obi, same white socks, and sandals only beige and blue. She keeps the white flowers and headband. But that's just me, ignore it if you will.
Personality: Can be a mix or separate to make it a natural change, probably inspired by Gon.
A bonus, heads up on Akuatsu shipping. Atsushi sees Illumi in a part of Akutagawa, and I'd believe he wants to be on good terms relating to Akutagawa as well. (That can be ignored too.)
No BSD and HxH crossover explanation. Summarize your own conclusion.
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