OKAY SO. Azula, Ozai, and Ursa are all going to be affected together (I'll be putting up a similar post in Ursa's journal soon), and the question we're going from is: What if Ursa had never been banished?
After Azulon's death, Ozai proposed a cover-up as opposed to a PR-necessitated banishment, and so they made it look as if he had died of natural causes. Ursa was therefore able to prevent the Zuko/Ozai Agni Kai a few years later, and Zuko was never scarred or banished himself. At this point in time, Ozai is still Fire Lord, and his wife and both children are still living with him in the capital.
Bottom line? Azula will be more like she was before she started to be at odds with Ozai. She won't be perfectly normal (er, normal for her) and in control because Ty Lee disappearing will still have been just as big a blow as it was in regular DDD!verse, but it'll be less of a "just one more thing on top of everything else" scenario, and she'll have an easier time of repressing pretending that it doesn't matter. She won't remember anything that came about because of Ozai's assassins (such as staying at Cuddy's house), and any conversations/viruses/relationships will be altered accordingly.
If anyone has any questions about how the virus will change how she sees your characters/their past interactions/etc., feel free to ask me here! Or you can just wait for the virus to start and find out for yourself. ;D