I LOVE not driving!!! And I used to LOVE driving... In the US I've lived in the country (where you had to have a car) and LA (where driving sucks but still most people have cars).
I love being able to walk anywhere I need to go, and I've just started talking the bus too, which has been super easy.
Trust me, it may be weird to you at first, but you'll love being able to walk everywhere, or take the Tube. IMHO, the Tube is one of the best public transport options I've ever been on (including NYC subway, Atlanta MARTA, etc etc). I also think it's by far one of the cleanest.
Public transport is much more convenient here also, especially in London, since towns and cities tend to be so condensed I've never really had a long bus ride, subway ride, etc. When I lived in LA it would take an hour to get to work that was 8 miles away because everything was so spread out.
You will be fine.