>I think Wykehamists is the word for people who went to the school


Quite correct - named after a clerical gentleman of that name. He was, I believe, a headmaster of some repute. (He left before my day)
A resident of the town, rather than the school would be called, I suppose, a Wincastrian. "Caster" and "Chester" are interchangeable, meaning "castle"- usually denotes a town where there was a Roman fort in the good old days of the Empire.
However, people from Newcastle are not New castrians - they are Geordies.