We're way up in the East Anglian no-mans-land (Norwich) and have managed without a car for 9 months now. The buses are great, there are trains to London (and from there you can go anywhere). We even splurge on things like taxis because it's *still* a heck of a lot cheaper than having to pay for things like cars, insurance, parking, petrol, etc.
Dh is working on getting his license here, but I'm just as happy to go without.
You will be more than fine without a car in London.