Is it possible to squeeze into a 3 bedroom? If so I would suggest a look at Greenwich. My wife and I used to live in central London in Marylebone (just North of Oxford St) but soon after she got pregnant we decided to move to a more child friendly neighbourhood. We visited Greenwich and fell in love with it. Although it’s close to the city it has a village feel (local butcher, cheese shops, pubs, etc). We moved into a rental but less than a year later decided to buy and make it a more permanent stay.
We know most of our neighbours on the street and we often baby-sit each other’s kids. Most times when we go to the park we run into families we know and chat for a bit. It is easier for us to meet people though as my wife isn’t working right now so she meets up with friends on weekdays.
Other local attractions are Greenwich Market, the Observatory, the newly opened Planetarium (opened in May), Naval College (once a Royal palace), and hopefully we will get the Cutty Sark back someday. We just took a boat cruise from Greenwich Pier up to Parliament last weekend for something different to do, our boy loved it. All these are within walking distance to anyone in West Greenwich. These make for a very busy/touristy town centre but it is easily escaped in the residential areas.
As far as commuting, PUBLISHED train times to Cannon Street (City) and Charing Cross (West London) are 13 minutes and 18 minutes, respectively. Beware published times no matter where you live though, as trains/tubes are always slower during rush hour due to congestion. I just checked on tfl.gov.uk for travel times to Oxford Circus and it gave 30-35 minutes for a time, which means 45 minutes due to delays.
I’m not sure about schools for older children as I have not researched it yet, but there is a good local school for our boy to attend when he reaches school age.
When we moved to Greenwich in 2004 we got a 3 bedroom on a nice street for £1450 per month. I think rents have gone up since then, I just did a search and prices in the nicer areas are higher now. 4 beds are harder to find and you would have to pay a fair amount for those.
Anyway, not even sure if you want to be on this side of town but thought I’d throw the idea out there.