If you're getting by hardly ever driving in a city in the US, I think you'll find it even easier in the UK. Most everyday shops you need will probably be in walking distance, plus I agree with what others said about the tube in London being the simplest subway system I've ever travelled on, and I've travelled in the US, UK, Germany, and France. Plus there are lots of buses.
Your American license will be valid for 12 months after arriving, so if you do want to practice driving to see what it'll be like you'll be able to on that. Also getting a provisional (learner's permit) UK license is very simple. Takes about 3 weeks and around £50. It's useful having UK id to carry around that's not as valuable as a passport, plus it's good for 10 years. Since you mentioned in another post that your mum has been living in the UK, if she's been fully licensed for more than 3 years then even after a year when your US license expires, you'll be able to practice on your provisional as long as she's in the car with you. Otherwise you could book lessons.
We live in a small town and got by for ages without a car. We finally got one because DH's parents live in the country (only about 4 miles away...) where public transportation is very sparse. We also use it for meeting friends at pubs or restaurants that are more than a mile away from where we live, or for fun country drives sometimes. But in general we get our groceries delivered, can walk to work and the shops, can take the train or bus to farther away places, and only use the car for meeting friends or family who don't live in towns or cities.