Bills will depend on the size of your house, obviously. But you should be able to get dual fuel on a 3-bed house for under £100pm, as long as you're not running the heating constantly.
Food: You can feed a family of 5 for £400 a month, if you put your mind to it, including household stuff. It's possibly cheaper in London than in the sticks because you have access to a greater range of supermarkets and shops, though obv you don't get the very cheap deals you get in the out of town supermarkets.
Transport: You don't need a car in London, you really don't. Not having a car is the real saving, as it will cost more to insure in London, plus you have to pay for residents' parking in most places, not to mention parking in places you drive to, and congestion charge. Much easier not to bother. If you really need a car, you can join one of the car clubs, zip car or Streetcar, or hire one for a few days. All the big supermarkets in central London have arrangements with minicabs, so if you want to do a really big shop once a month you can bring it home that way, otherwise use a delivery service or buy a little trolley for topup shops.
Public transport: depends on what your commute to work is like. Buses are quite cheap (and free for school-age kids), tube gets more expensive, train more so again. If you can walk or bike, then that's practically free.
After-school activities: depends a bit what you do, but most are in the region of £70-80 a term for eg. drama, ballet or swimming. One-to-one music lessons are more, activities organised by the school generally less. There are loads of fantastic activities you can do for free at weekends and after after school as well, plus parks, museums, etc etc.