What zone? Camerwell is good, further out around Bushy park. Run down but surely on the up due to location and regeneration and good investment include elephant and castle, a bit carp but people don't realise you can walk to south bank in 10 minutes, same with Vauxhall.
Tooting is good value and attracting hound professionals looking for good value.
Further out previous holes now attracting artists looking for cheap studio space so of course the gentrification that follows include New Cross, Peckham and around there.
Also investigate the new train line linking Croydon and Crystal Palace meaning loads of new flats and good links for London.
Wandsworth and Battersea have good vibes and are more established areas rather than up and coming.
Pimlico and Victoria would also be worth checking to see if viable in budget.
Lastly are you sure not north of the river?! I love where I am around Kensal Rise / Queens Park.