In the London region and its suburbs and outer London, I'd say in my opinion the answer is a little bit of both...you don't need a car to get around, you CAN get by, but a lot of things about daily life is more convenient if you do have one.
There are plenty of buses, tube stations and railway to get you to most places you'd want to go, certainly within the heart of London ( and you actively don't want to actually drive in central London if you can help it; traffic, congestion charge, etc).
But on the edges and just outside London which is where I live, I find that there is frequently a store, shopping centre or some other thing I'd like to go to but the public tranportation (still covered by London Transport) to that particular spot is either minimal or not conveniently close, especially if it's a heavy or bulky item I'm wanting to buy -- I recently walked about a mile back to the nearest tube with a 15 lb item I then lugged on the train with me. That wasn't fun, and for things like that, you need the comfort and convenience of a car even though technically the public transportation IS there.....
My verdict: even in greater London, a car is a nice thing to have for those awkward locations and awkward missions.