They do not necessarily hold photography exhibitions, but an interesting place that holds up pop-up exhibitions of various different artist’s works is Chatsworth House. On one of the last times I visited, they showed many different and unusually designed chairs in the rooms the public are allowed to go in. The public were allowed to sit and touch and photograph the chairs to their hearts content (most historic places do not let you near anything!) which I really liked. Holding an exhibition inside of a stately home is quite unusual, especially when you are allowed to actually touch them and go near them. I also like how they are spread out like a treasure hunt around the house and gardens, making the experience more interesting.
Here is a link to their recent exhibition:
www.chatsworth.org/attractions-and-events/events/never-a-bore
and some photos of the artwork from when I have been: