My wife and I visited the UK a few years ago.
In the London area, lodging seems to get significantly more expensive the closer you get to the city center. We found a hotel a short walk from the end of one of the Tube lines (Walthamstow Central station, to be precise) that was quite reasonable in price and still allowed us to get to tourist attractions pretty quickly.
There are lots of really nice museums throughout the country that have no admission charges.
We didn't need to rent a car while we were there because we stuck to cities. We spent four days in London, a day in Leeds, a day or two in York, two days in Edinburgh, and a day in Liverpool, where we then caught a ferry to Ireland. We did some of the intercity trips by bus, some by train, whichever was cheapest.