Saikyō Hīrō no Kōkō
Translation: High School of the Strongest Heroes
(The grand name is seen as a ironically hypocritical by the other hero schools since this one is considered the worst)
(UA Hero school rejects)
Izuku WASN'T able to get into UA (whether he's got One-For-All or not is irrelevant in this situation) because he'd failed to get even a single Villain point and, (if he does have One-For-All) his Quirk was deemed too dangerous for himself and those around him to be feasible as a Hero. Instead, Izuku gets shuffled off to some low-bar hero school instead. So, he will be starting in the absolute worst hero school in the country and having to fight tooth and nail to get even half the respect and prestige they'd have gotten from UA. An uphill battle every step of the way during his quest to become the next #1 Hero of the World.
Much to Izuku's consternation, Katsuki Bakugo was successfully accepted into UA Academy despite having had 0 Rescue points. The teachers felt that they could instill some much-needed empathy into him as he matured.
On a side note, for those of his UA classmates that could've 'followed' him to this junkyard of a high school I feel could be people like Ashido, Uraraka, Sato, Kaminari, Mineta, and/or Kirishima. The kids who were either among the lower half of the class rankings, or came from poorer families who couldn't afford the education costs of UA, or just didn't score that high on the entrance exam.