You can get info on last few years catchment areas by looking at the councils websites for primary school applications.
Shortest commute with good school would be from Mudchute DLR. There are some cute dockers cottages near the DLR station/ park/ Canary Wharf college (which is outstanding and has a class size limit of 20 - neighbours daughter goes there and loves it). Area is mixed, nicer places near the river.
There is a lot more going on in Greenwich, plus the park is much nicer and bigger, tho it does get over-run with tourists.
Blackheath is lovely, though the area is very mixed and most well off people send their children to private schools, so state schools are more diverse.
Higher Green has a few outstanding schools, though the area is very overpriced in my opinion (it has undergone rapid gentrification recently). Less affluent than Greenwich / Blackheath, and beside Lewisham and Catford which are not good areas.
Wanstead is nice and has quick links to Canary Wharf and the city, and has a lovely forest park. However not as nice as Greenwich / Blackheath imo.
Bromley South is also nice, tho harder commute to Canary Wharf (trains go to Victoria) tho they are due to relocate La Fontaine Academy there, which is a bilingual French new free school, which I hear good things about.