Most public buildings don’t allow non-assistance dogs to enter, but many welcome dogs in the grounds. Sometimes enjoying the architecture from the outside can be just as interesting.
Both English Heritage and the National Trust have online lists of their dog-friendly properties. (For the National Trust, select ‘refine your search’ and click the box to tick ‘Dog walking’ before clicking ‘Apply’ to see the results).
There are lots of dog-friendly public gardens across the country, although many might restrict which areas you can enter and also require dogs to be on leads. Don’t bank on there being a poop bin either, so make sure you take plenty of bags and a means to hygienically carry the used ones! A nicely laid-out listing with map view is available online via the Britain’s Finest website. Although many organisations are efficient at keeping their websites up to date, it is always worth contacting the place you plan to visit, just to check nothing has changed.