Adachi had joined Blackwatch but with the dissolution of Overwatch joins the police force instead. He doesn't agree with Talon or their mission goals but he still kinda wants to bully people, you know? He gets passed up time and time again, gets himself into hot water, gets sent out to Inaba.
Yu is the most interesting change. I have it headcanoned that he and his parents were in a car crash early on in his life. He lost most of his lower body and his left side is covered in burns and cuts [the ones on his face have been repaired for the most part and he hides them even further with synthetic skin] He has almost no real organs in his body to speak of [save for his heart]. His legs and arms have been replaced by prosthetics. And in order to keep the synthetic organs in good working order, his blood has been replaced with something that works similarly to it but is green [like Genji] He was living with his caretaker who happened to be an omnic so he was really sensitive to the violence that happens to omnics and in order to try and save his caretaker, he lost his right arm. They died and after the funeral that he holds for them gets shipped out to Inaba. OH! And to pay for all of the machinery inside of him, he uses protoypes from Vishkar Corporation. He's usually in a lot of pain, so he has a cane that he uses. Not a lot of people believe he actually NEEDS it though and so kinda gets bullied about it. [He doesn’t need it as much as he leads people to believe however.]
The omnic that Izanami built [codenamed Marie] greets them at the Moel Gas station. She marks them.
Unbeknowst to Namatame and Adachi, they get followed by omnics [codenamed Kanon and Ameno-Sagiri respectively]
Yu gets an omnic too. One who's decided the best way to spy on Yu would be to dress as a mascot character. Teddie is quickly found out and befriended.
The events of Persona 4 play out mostly the same, save for a few critical differences. The shadows are all omnics, the fog is metaphorical instead of literal, there is no TV world, and in order to hide the sword, Yu has a cane sword which is badass and cool.
Oh, and all of the 'shadow selves' from Persona 4? Omnics built to resemble the people they're mocking. They know a surprising amount of information about the people that they're trying to break down.
Even Teddie gets a shadow omnic. Which is just depressing when you think about it. [headcanon that he suffered a knock to the head and so some of his programming got messed up so he can’t quite remember what he was supposed to be following Yu around for.]
But sometime in May [normally when P4 ends, Yu doesn't have any place to go back to because he is literally living with his uncle and cousin until he moves out] Yu finally puzzles out what's been bothering him so much- the omnics aren't doing this on purpose. There's a God AI on the loose.
Everyone goes after the God AI. Everyone nearly dies because of it. But when confronted with his choices, Yu stands by them and his friends and shuts Izanami down. The AI isn't at all upset about this. She just congratulates him. And what little she could fit into an omnic gets transferred over into Marie [now Mariko]
Everyone's wary at first, but they do warm up to her since she does provide a lot of useful information and a source of entertainment [since she's never had the reasons behind why people do things, only that they do]
Three years after that, Overwatch gets reformed
Everyone wants to go but Yu vetoes them gently. Until omnics start attacking in Inaba once more, that is. Why? No one knows!
Hanzo, Genji, Zenyatta, Mercy, Phara, and Symmetra are the ones to find them because they were getting overwhelmed until Yu stepped in. And where Yu steps in, the rest follow in a really cool, really epic fight scene
They wind up in Yu's house [as they're wont to do] when both Symmetra and Yu start feeling pains in their prosthetics. Symmetra's able to take her's off.
Yu's small intestines shred themselves inside his body. Mercy- thankfully- was able to stabilize him enough to get him to the hospital where he gets hooked up to a lot of machines. Why is Vishkar Corp turning their products against their customers? WHO KNOWS! It's time to find out though.
And that's how the Investigation Team becomes an unofficial part of Overwatch.