On the contrary, SW London has some of the largest green spaces in all of London: Richmond Park (largest Royal Park in London), Wimbledon Common, Barnes Common, Kew Gardens, Old Deer Park and a little further out in Twickenham: Bushy Park. The river itself has a tow path that you can cycle and walk/run along. Seriously, look at London on Google Maps in satellite and you'll see that SW London is in fact the greenest area of town. It just happens to be a bit further out from town. As you can imagine, the closer you are, the less green space there is.
If you are looking for the green space / back gardens combo, look around Richmond, Twickenham and Wimbledon. All are incredibly family friendly and I believe you'd be able to find homes with back gardens very easily, though I can't speak to the schools. Your husband will be mainly relying on rail links to get into town which is the downside of the SW (even though Richmond and Wimbledon have the tube, it's not quicker), but it shouldn't take him more than 40 minutes as both those locations have excellent high speed links into town to compensate (with their delay headaches, of course, as any rail line does in London).
Sounds like you weren't looking that far out, and I don't mean to confuse the issue further! Do you think your relocation people would send you an example of properties they could show in your chosen areas to help you narrow down your search? At the moment Right Move and those types of sites are not really up to date with their inventory.