In England, private schools such as Eton are called public schools, weirdly enough (to this American).

Jeeves would have to have been largely self-educated since he was in service from such a young age. University was out of the question. He might have gone to school off and on. But I think he was mostly an autodidact. Occasionally I've wondered if he resented a fool like Bertie getting an Eton and Oxford education.

Elementary school is primary school; high school is secondary school.