I would look at:
Bexleyheath
Bexley
Bromley
I know the first two very well - excellent schools and a good variety of shops and transport links, and lots of green spaces and parks. Also, the A2 is close, meaning you can drive in to Kent for plenty of countryside, history and beaches. You can even surf at at least one of the Kent beaches! The train links are good too if you don’t have access to a car.
£1500 would get you a 3 bedroom house with garden. You can find houses costing less than this with 3 bedrooms, and obviously 2 bedroom houses too. The area around Danson Park and William Morris’s Red House are very good (park, Red House, gym, pub, and schools). The town centre is around a 15 minute walk from there too (shops, supermarkets, transport, cinema, bowling, restaurants & cafés.
Bromley has a great town centre, and fast trains into London Victoria only take 15 minutes from Bromley South, but I don’t know about the schools etc. I only really know the shops!
As you could go up to £2000 per month, you could afford somewhere on the borders of Greenwich or Blackheath, which are lovely. Look at Charlton and Kidbrooke for this (but only areas on the borders).
I personally favour, and recommend Bexleyheath for the reasons above, as well as it being cheaper than near to Greenwich and Blackheath. These places are only a short train or bus journey away from Bexleyheath, or a short car journey (15 mins approx) on the A2.