The Underground. Commonly called the Tube, it is a subway system run by Transport for London. London invented the subway so naturally they have the world's largest system. If you're able bodied, it goes just about everywhere. If you're in a wheelchair, you can still get to a lot of destinations. Heathrow, Westminster, Tower of London, and Waterloo are just a few of the accessible destinations. Picadilly Circus and Hyde Park are a couple of examples where there is no accessible station.
Each station has a barrier-free map available, also available online, that shows all accessible stations. Make sure you look at the back of the map, page 2 of the link above, where each station's gap is listed. "Mind the Gap" is a very famous warning given to Tube riders in each station. It is the difference in height between the platform and the train's floor. It can range anywhere from nothing to an impossible 12 inches, so study this part of the map too. You can find a station with ramps or lifts all the way down to the platform and then be stuck because the gap is too large to overcome.
Full cash fares start at £4.00 (about $8 - told you it's expensive!) and go up to £10.80 for the longest journeys. A peak-time day pass goes for £7.20 to £18.00. If you can ride off-peak, it's £5.60 to £14.20. Either option will save you major cash. Passes are also good on the buses. 7-day passes are also available at approximately 3 times the peak day-pass cost.
There is no official way for disabled visitors to get a discount (you can apply for a Freedom Pass but the process seems way too cumbersome for a visitor) but, unnofficially, we've found that the attendants in the Tube stations will let wheelchair users through the turnstyles at no charge...your mileage may vary.
Buses. Also run by TfL, the buses go everywhere and are completely accessible with the exception of two lines, 9 and 15, which are run as "heritage" lines with historic double deckers that are not accessible. The modern double decker buses are accessible. Wheelchairs enter through the back door which has a ramp that is only able to be deployed when the front door is closed (here are instructions). Make sure the driver sees you, knows that you want to board, and then wait at the back door for the ramp. The doors will close but that does not mean the driver is leaving you, the ramp will then be deployed. There is a spot adjacent to the door where you need to move your chair to and sit backward against the board in the space. There are no tie-downs like you'd see on American buses but the board keeps you from moving towards the front of the bus and a bar holds the side of your chair.
Fare is £2.00. Disabled riders are free, however your companions must pay the fare. Fare inspectors frequently show up on buses and demand immediate payment of the £50.00 fine if caught without the proper fare (if you truly do not have the money, policy is to give you 21 days to pay...your fine is cut in half if you pay within that time).
National Rail is the umbrella company that oversees the many train companies working in Britain. Trains make it easy to visit other towns, such as Bath, for a day trip. You can look up fares and timetables at that link. One big hurdle for disabled passengers is that you need to give 24 hours notice for guaranteed assistance at the station. Doesn't sound bad but we thought we had done that, only to find out in Bath that our notice was only good for the training going there, not coming back. We were almost denied boarding. This compares to only 30 minutes notice for trains in France, a much more accomodating system in my mind.
Water Taxi. Accessible boats ply the River Thames for a more leisurely way to get about. There are two main companies, Thames Clippers and Thames Executive Charters. Click on the links for more information.