I was a caretaker for my handicapped mother for many years so perhaps I have a heightened awareness of mobility and access issues, but if you have any kind of issues or restrictions be aware that you will run into issues in London.
Not in newer facilities, which by law must be accessible, but in older buildings of which there are many many. One trick I noticed is that oftentimes a pub will have a dining room, which is found at the top of a long narrow staircase. And then, if you make it that far and need to use the loo, you will either have to go all the way back downstairs or, in a novel twist, will have to go down a half staircase to a landing with restrooms.
This demi-floor restroom situation was also found in the Victoria & Albert museum. And while there must have been elevators in that museum, when we asked the guides at the door they got into a spirited but polite discussion about which elevators were open, and which were closed, and which were for members or staff only, and which stopped at which floor. In other words, to get to the top floors you will be climbing many stairs.
Also each Tube station was built at a different time to a different pattern, and there might be odd little diversions where you'll be walking along and come to a small staircase of 3 or 4 steps to a landing and then 3 or 4 steps down. Which could also do you in if you're dragging a suitcase. At Paddington Station the loos were also downstairs, and there were signs directing the handicapped to other bathrooms which were apparently all the way across the station.
The most interesting station I visited was Covent Garden. Apparently this is so deep that the only sane way to the surface is via elevators, of which there were 4 or 5 each of which carries 50 persons. So in other words they're big enough for a car. There are warning signs at the stairways that the climb up is equivalent to a 5 story building and you really don't want to do it.