Hello Marc,
Greenwich is nice, as is neighbouring Blackheath, although may be a little expensive. Seems to be lots of green space in that corner of London, with the heath and Greenwich Park, transport links are good as well.

Never rated Wimbledon, neighbouring Putney is quite nice, but again your paying through the nose for what you get (seem to recall paying ?900pcm for a one bed basement flat not so long back!) and its not that great.

Might be worth looking at St Albans, its north of London granted, but has excellent transport links via Thames Link trains into the City and key hubs such as Kings Cross where you can hop on the tube. Its an old Roman City so quite nice, plus you get a lot more for your money than in central London. Although it is outside of Zone 6 so you will pay more for the commute, which will have to be factored into the savings on rent.
Keep in touch