The UA students and faculty from the Hero Class: Civil Warfare universe weren't transported into the canon universe, but in one generated by the change their very arrival would cause in the latter. So while the characters we know and love went on with their lives, completely unaware and unaffected, doubles of themselves came into a new existence influenced by the arrival of their alternate selves. By creating this temporary "branch", reality itself prevents any potential disruption caused by what is still the arrival of beings from an universe to another, even if they're almost identical ones.
The moment the Villain Team students and their teachers return to their original world, the others will not just forget everything that happened: they'll blink out of existence the same way they came into it, an universe that lasted only a couple weeks, just a microscopic flicker in the infinite sea of possible universes.
Granted, this interpretation would make the ending, with the Villain Team remembering the whole experience while their counterpants will simply disappear, much more bitter than sweet.