Finding a rental with a dog is difficult. My husband and I went to estate agents and when they first looked at us, 40s, professional, no kids, they were so enthusiatic about finding us a place. Then I mentioned I had a dog, and they looked at us like I had just said "oh, make sure the garden is really big because we need a place to bury the bodies." It was frustrating, but eventually we found a nice place, in a nice area, and about a 2 minute walk to a nice off leash dog park.
Once you find the house, I think you will find the UK very dog friendly. I think it is much more friendly than Los Angeles, where we lived before here.
There must be 5 parks, all within 15 minutes walk that allow dogs off leash near us. But as other posters have mentioned, some people need to be a little more vigilant about cleaning up after their dogs.