opening-times.co.uk doesn't looks like it carries museum opening hours. Actually a semweb data source of visitor attractions and their opening hours would be a very valuable resource. I don't think one exists.

I wouldn't be surprised to find that foursquare check-in's is a very poor proxy for this kind of venue. For bars/clubs and coffee shops maybe, but you're measuring with the wrong stick. Having said that, it does seem to confirm common sense (that most places are busiest on the weekends and mid-week is quiet).