I am looking for a humanities program (English lit, theatre, art history or museum studies) that I can do part-time distance, or online. I have looked online a lot but cannot find any bricks and mortar uni that fills the bill. Have looked at OU, but do they have a Ph.D? I am an American interested in a UK degree, so I am also looking for scholarships since I am not EU.
Thanks in advance for any info.

http://www.open.ac.uk/Arts/research/resgroups.htm

As an example, this was very easy to find. I am surprised that you did not find this. It takes very little effort to track down such information using a search engine such as google and a little navigation through a web site.

That said, I hope this helps answer in part at least one of your questions.

The University of Leicester have distance MAs in museum and related studies so it might be worth checking re PhDs.

The subjects you describe are hugely diverse! Are you properly focussed on your field? Although there are relationships, english lit and art history, for example, are completely different disciplines!

Perhaps you need to identify your subject first, then the field within it.