All non catholic schools (and the one Jewish primary and I think there is one Church of Scotland school) are by definition "non-denominational" 
That's been the case since 1928(?) when the Education Act brought all schools into state control. The Catholic Church didn't want to give up control - hence the parallel system, but the Church of Scotland did give up its schools (with maybe one exception that I have a vague recollection of
)
All schools were/are still supposed to reflect the Christian "established" church (not the correct word in Scotland as we don't have an established church but the "state" religion is Protestant): that was why schools still had to have assemblies that were "broadly Christian - although in practice, at least at my schools, that amounted to trips once a term to to services at the local Church of Scotland (at least Christmas and Easter - I don't remember a summer visit).
Even ds' secondary school still has Easter and Christmas services at the local Church of Scotland but with such a high Muslim (and other) population, it doesn't enforce attendance - and doesn't seem to even do registration on those occasions, so I never forced ds to go. Come to think of it, with 60% of its population Muslim, ds' primary school didn't even go to any services - but it did have Nativity, Eid and Diwali Assemblies which incorporated education for both pupils and parents about the respective religions 
I would hope that the headmaster in the "posh" area of Bearsden who was explicitly prejudiced against Catholics was a dinosaur - and it was 50 years ago (
) - but I do remember at my last workplace (a nice office environment, full of educated people) still a few dinosaurs who demonstrated prejudice
I was
that it could still exist.