OP, are you sure that both Wimbledon/Southfields and Dulwich would be good choices for commuting? Where would you need to commute to?
Wimbledon is one of the best-connected areas of south London, with trains to Clapham Junction (from which you can change and go to Victoria) and Waterloo every 3 minutes or so at rush hour. Plus the tube.
Southfields has only the tube (same line as Wimbledon), but that’s not exactly the most frequent line nor the fastest; overground trains from Wimbledon to Waterloo are typically more frequent than tube trains.
What do you mean by Dulwich? The Dulwich Village is served by the North Dulwich station only. East Dulwich by the East Dulwich station, plus Denmark Hill isn’t too far, either.
A big problem with Dulwich is that it’s served by Southern Rail, and that line was an absolute disaster over the last couple of years. Lots of strikes and cancellations. One summer there was one train per hour at peak time! It seems to have got better recently, but transport links in Wimbledon are still much better. The difference is not so much that, when all goes well, you must wait 7 more minutes for the train to Dulwich; it’s that, when things don’t go well, commuting from Dulwich may take much much much much longer.
Have you also considered other parts of Wandsworth? Balham is well connected with very frequent trains + tube (for some reason, Southern Rail didn’t neglect Balham as much as it did Dulwich). The whole area between the commons (between Wandsworth and Clapham Common) is very popular with families and with good schools; depending on the exact location, you could commute from Clapham Junction or from Balham or Clapham South (but the tube from Clapham South can be incredibly crowded). Earlsfield is halfway between Wimbledon and Clapham Junction (one stop from each) and also has good primary schools; cheaper houses because there isn’t much around there.
One piece of advice: take feedback with truckloads of salt. Many Londoners will tell you that wherever they live is best, will not admit their commute is hellish (I have heard people telling me that getting into north London from Bromley is ‘convenient’ – I kid you not) and will want to convince you that the school of their little Johnny is an Oxford for little children.
Have you familiarised yourself with the criteria of religious schools, if they are an option? The idea that all taxpayers fund state schools which are only accessible to some religious minority is IMHO unworthy of a civilised society, but, if you don’t find that as aggravating as I do, one religious schools may in fact be a good choice.
You need to think strategically of a location from where to apply to both good primaries and good secondaries, right? In Dulwich there are two good Charter schools (I think they’re part of the same trust), one near North Dulwich, one near East Dulwich, both with some primaries nearby.
In Wandsworth, Graveney is considered a very good secondary, and part of its intake is by merit. I wouldn’t move near there (so as to get in by distance), though, because transport isn’t great from there.
Other options in Wandsworth might be Balham: Chestnut Grove is considered a good secondary and has some primaries nearby, but Graveney seems to have a better reputation (especially for kids who want to study science at uni).
Or East Putney, with the Ashcroft Technology as secondary.
Or between the commons (Ark Bollingbroke as secondary, Honeywell as primary).
I have no direct knowledge of these schools, by the way.