I can't believe no one has mentioned Ealing yet! It is a super family friendly borough - green and self contained with plenty of shops, cafes (chain and individually owned), restaurants, bars, loads of stuff for kids to do and very friendly. Ealing Broadway is a 10 minute direct train ride to Paddington as well plus a few tube stops from the fab Westfield shopping mall and is on the central, Picadilly and district line as well as Paddington mainline rail service and loads of bus routes. Also close to the motorway for getting away easily for weekend breaks to the coast - Devon, Somerset, Wales all within a few hours drive and Cornwall a five hour or so drive away.
Ealing also has a fantastic family club - the 135 group (named after the two main Eaing postcodes W5 and W13) for parents of under 5s www.ealing135.org.uk/ - which organises day trips, park picnics, open houses (drop in coffee mornings/ afternoons), Halloween/ Easter/ summer/ Christmas parties, playgroups, BBQs, nearly new sales, support groups and many, many other daily activities.
I've lived here for 10 years (originally from Wales) and have a 2.5 year old daughter and 6 month old son - can't recommend it enough as it's so friendly. I have made dozens of mum friends. Chelsea, Kensington, Notting Hill are posher but you won't find them so friendly - they are lovely places though and worth checking out - as is Chiswick (which would be my second choice of place to live with children as it's by the river. Ealing wins on parks though including the lovely Lammas park with a children's play centre (with indoor play hut all free), Walpole park with its great cafe and Bunny Park in Hanwell with it's country feel, maze, and animal area (peacocks, monkeys, wallabies, pigs amongst others - no entrance charge).
A new venture started just the other week which runs classes every week day for babies and toddlers called Pitter Patter the Hub for Bubs:
www.westlondonmum.co.uk/pitter-patter%E2%80%94the-hub-for-bubs-ealing/
www.pitterpatterhub.com/
Pitzhangar and Northfields are particularly nice areas of Ealing with lovely shops and restaurants, bakeries, libraries etc but South Ealing and Ealing Broadway have nice areas too (for the latter particularly round Walpole and Lammas parks).
It's not as central as some of the other places mentioned such as Notting Hill, Chelsea and Kensingon - which are all lovely areas too - but probably better connected to Paddington with a train ever 10 - 15 minutes which takes 10 minutes (also direct mainline train and tube to Heathrow - the former taking 10 mins or so and the later 15/ 20 mins) but it is super friendly, a great community, green with loads of parks, a common and other open spaces, a yearly Ealing Festival in the park which offers Blues, Jazz, Opera amongst other things and extremely multi cultural (I have friends here from every continent except Antartica! Know people from most countries in Europe including Iceland and Andorra and also friends from Brazil, Mexico, Columbian, kenya, Nigeria, South Africa, Australia, New Zealand, China, Korea, Japan, Russia - you get the picture!
It is also a lot cheaper than those more central places and £1,000 a week will get you a 5 bedroom house with a decent sized garden in a nice area near good schools (both private and state) and nurseries.
Anyway there's my case :) - as you can tell I'm quite a fan of my borough and so feel free to contact me for any more info.