And more specifically which stations would you run into no problems being at 24 hours a day?
Presumably the thameslink 24hr stations, Derby, Reading, Manchester Airport, Manchester picadilly. Where else?
Pretty much all smaller ones in that there's usually no way to physically close them and probably nobody watching the CCTV.
MKC is open all night on weeknights due to there being a service in one direction or the other all night*, same as Bletchley - both of these can only be accessed via indoor ticket halls with no side entrance. There are private security staff overnight at both, though, so you might be questioned for hanging round if you don't have evidence of planning to travel. I have, that said, used MKC as a late night taxi waiting room before. Both get locked up for Christmas and Boxing Day, though. They may lock them up on Saturday-onto-Sunday due to the longer gap, I'm not sure.
* After a weeknight, one 4-car set doesn't "go to bed" - the 0134 from Euston forms the 0315ish from MKC, so there's activity of some kind at MKC and BLY every hour all night.
Euston, Paddington, Liverpool Street..?
Euston is locked up between the 0134 departure (after weekdays) or the 0200 RRB (after Saturdays) and reopened for the first train. I've not experienced it (came very close to it once!) but the departure board says so.