We were in England for July, staying in Central London at the tail end. It is an eminently walkable city.
I was up and out by 9 am and managed to get in a couple of two to three hour walks in before my daughter roused herself. One day, I went to The Bank of England to change sone old notes and the walk from Bloomsbury covered Greys Inn, the courts, the Royal Exchange, St Mary le Bow and a few other churches, The Royal Exchange, Daunt Books, St Paul's Cathedral walk by, a sojourn in the grounds of Greys Inn to read my book.
The second day: Bloomsbury to Trafalgar Square, Pall Mall, a walk by Buckingham Palace, Green Park, walk by The Ritz, a stop at Floris, stop for tea at Waterstone's, walk through Soho and back to Bloomsbury. That day we bussed to Hampstead for lunch and a walk on the Heath to Kenwood House, whose cafe unexpectedly closed early at 4:00 pm. I did 30,000 steps according to my watch.
I second the suggestion of Bletchley Park. If you are a real computer/math geek, the National Computing Museum is next to it. We went to both last year on a short break to Milton Keynes for another of our geeky interests: Take That. Another story.