I would have a look at Balham (SW12 postcode). It's on the Northern Line, so easy to get into The City (original financial district) and Canary Wharf (more modern financial district).
You won't have any problems getting a 2 bedroom flat for your budget - you might even be able to get a 3 bedroom flat. Some won't have gardens, but those that do will probably have small 'London, postage-stamp' gardens. But, there are three Commons (large parks) within walking distance : Clapham Common to the North East of Clapham South tube station, Wandsworth Common to the North West of Balham town centre and Tooting Bec Common to the South of Balham (Tooting Bec Common also has an outdoor Lido - great in the hot Summer months!).
You could also look at some roads in SW17 (between Balham station and Tooting Bec Station (but I wouldn't head any further west, if only because of your walk to the tube) or look at roads in SW11 (but only on the south side of Broomwood Road running towards Nightingale Lane, again because of the walk to a tube station). I think a 10 minute walk to a station is great, 15 minutes OK and 20 minutes would, probably, be the maximum edge of your walk.
So, you have the choice of three Northern Line stations - Tooting Bec, then Balham then Clapham South.
It is a very family friendly area and many people have moved to the area because they work in The City/Canary Wharf. Many families are international. There are loads of schools - private (fee-paying) and state (non-fee paying). Unfortunately I can't help with recommendations, but perhaps a 'local' might give you some pointers.
There is an area known locally as The Nightingale Triangle. Have a look there (see one of the maps, it's roughly between Clapham South, Wandsworth Common and Balham stations). Or just to the south of Wandsworth Common, between Bellevue Road and crossing over Balham High Road (into a lovely area called The Heaver Estate).
The majority of the architecture is Victorian - and some Edwardian - so rather attractive! (and don't be put off by the name 'Heaver Estate'; it's not a modern, awful housing estate, but was built by the Heaver brothers in the 1800s and is well-known for it's excellent Victorian architecture!)
If you do consider this area, there's a good local forum called Nappy Valley Net nappyvalleynet.com, which is a bit like MN - but just local to the area.