Agree that 'college' is not appropriate not only because it's usually used to denote a sixth form / FE college, but also because it's problematic for collegiate universities
'Sixth form college' is a uniquely British institution.
While there are colleges within American universities (College of Engineering, College of Liberal Arts and Sciences, College of Agricultural Sciences, etc) Americans use the term 'college' for university. When I was at university in Dublin in the mid 80s people in Ireland used that term too. This had nothing to do with the fact that the two biggest universities were University College Dublin and Trinity College Dublin - students at other universities that did not have the word 'College' in their names also used the term 'college'.
Americans also use the term 'school' for university/college. Some use the term School interchangeably with College in the name of a College within a university and if you google (for eg.) 'school of liberal arts university of minnesota' your result will be the College of Liberal Arts at U Minn.
Your neighbours might ask, 'Where is Prudence going to school?' This would not mean, 'What high school will Prudence be attending?' 'Going-to-school' means 'going to university' (aka college). They might also ask, 'Where is Prudence going for school?' This means the same thing.
In a lot of cases there is no need to ask where anyone is going to high school - you go to the local public school.