We're in Zone 2 (Hampstead/Belsize Park area) with no car. Ironically, we chose this area precisely so that we COULD have a car. But it got put off because we had so much else to do at first, and as time goes on, it's looking less and less likely that we'll actually get one. My groceries are delivered, my kids' school is within walking distance as are 3 different High Streets, and it's easy to take public transportation around the area or into central London. Quite honestly, the reason that we haven't gotten a car yet is that as time passes and I get more and more accustomed to using my feet, public transportation and the odd taxi, the less I'm sure when I'd actually *use* a car (traffic is unbelievable and parking is so horrific that driving to do errands, etc. is actually LESS convenient than walking or taking public transportation most of the time).
The one thing that's making this all possible is the fact that my kids' school is 4 blocks away. We picked this home precisely because of its proximity to the school, even though we'd seen "nicer" places. If you're really serious about forgoing a car, I'd make sure that you'll have an easy school run (don't know how old your kids are, but presumably this will be an issue eventually, if not now).
One other thing -- there are a couple of services here where you can reserve "shared" cars parked in central locations -- either for a few hours or a few days. We've been looking into this as an option for larger shopping trips and trips out of town. You may want to investigate as well... they seem like a great inexpensive, convenient option for occasional car use. Check out Whizz Go and Street Car (don't know urls, but I'm sure you can Google...).
Good luck!
Good luck!