If you have no problems affording private education, then I would suggest Croydon - although this became part of the Greater London area back in the '60s, it is still essentially Surrey, and gets more picturesque the further south you go (Purley, Kenley, Reedham etc). You could get a 3 bedroom house within 10 minutes walk to East Croydon station for anything from £230k upwards, leaving you with plenty change left over to do it up. Alternatively, go for a property within a 5 minute walk of a tram stop, then hop on the tram to East Croydon Station.
East Croydon station has direct trains to London Bridge (15 minutes) of which there are about 8-10 an hour. There are also similarly numbered trains to London Victoria (16 minutes), Clapham Junction (10 minutes) and Gatwick Airport (15-20 minutes). You have the option of West Croydon station which has the Overground to Canada Water (25 minutes) with an easier change for the Jubilee line to Canary Wharf (one stop away) (I also work in Canary Wharf and travel there using this route - there's a lot to be said about always getting a seat). East Croydon station also has the tram (25 minutes to Wimbledon).
Alternatively, you can travel from the more picturesque parts (Purley station to London Bridge is 23 - 27 minutes; Sanderstead/Coulsdon South to London Bridge is 27 - 35 minutes direct or with a quick change at East Croydon station).
Primary schools
State schools
It's not great unless you live in
- the Park Hill/South Croydon area (Park Hill Primary, St Peters Primary - which despite its name is not a religious school); or
- Sanderstead/East Purley (Ridgeway, Atwood and Gresham primaries).
If you are religious, there's always
- Regina Coeli (Catholic) (South Croydon/Purley)
- Margaret Roper (Catholic) (Purley)
- Christ Church (CofE) (Purley)
Private schools
(for girls)
- Cumnor House
- Croydon High
(for boys)
(mixed)
Secondary Schools
State
This is where you run into problems unless you are religious or go private. Mixed, you have Archbishop Tenison (CofE). Girls, you have Coloma Convent (Catholic).
Private
More choice of good schools.
For boys (Whitgift, Trinity).
For girls (Old Palace of John Whitgift, Croydon High)
Mixed (Royal Russell)
Failing that, your child/ren will be older by then and can travel to school alone - this opens up a lot of London to you and all the choices that brings with it (there are kids in Croydon who commute to Westminster, St Pauls etc).
Grammar
There's Wallington for girls and Wallington for boys (though IIRC, these have no set catchment so there is intense competition for places from kids all across south London).
Poor Croydon gets a bad rep a lot of the time but you really can't beat it transport-wise with reasonable property prices to boot.