I'm pretty sure that when i was at school in the UK, there were specific school bus routes. They weren't operated by the schools or the local education authority, so i assume they were special services run by the local bus operators, using stock that would be used on other routes at other times.

I believe each route served multiple schools - kids from my school would share the bus with kids from other schools. Since we were the grammar school, full of nerds in garish purple blazers, and they were, well, not, this was sometimes a stressful experience.

It may be relevant that this was in Colchester, a town with numerous schools, some quite close to each other, whose catchment areas take in lots of small villages, where there isn't necessarily a good normal bus service.

Mind you, i got the (ordinary!) train, so i'm not sure of the detail.