Yes, it can be personal study. You don't need to be affiliated with an institution. Perhaps if you're asking to look at a Monet they might want a bit more information (Marcel Broodthaers isn't exactly a household name!), but if your hobby is painting and you want to study the brushstrokes I don't think they'd be able to turn you down - the whole point of these institutions is that they're paid for by the state, and by extension the taxes of its citizens.

Yeh that makes sense I guess. Conversely, a family of 4 turning up to get a private viewing doesn't seem in the spirit, even if it's their home-schooling art class.

I was curious with respect to places like the Tate that are really only partly state funded (30%) so I'd consider it reasonable for them to approve the most "worthy" 30% of requests.

Yeah, for sure - if you and some mates just want to gawp at a Picasso I think you'd get pretty short shrift!

I don’t know about museums but I do know that many public bodies in the UK will take enquiries from any member of the public, and will not automatically turn down requests - although such requests may go to the bottom of the priority queue if there are other requests of greater public interest.