OP, i grew up in Southampton, went to sixth form in Winchester and now live in SW London.
I won't bother covering Winchester as pps have said pretty much everything.
For Southampton- I think it gets a bit of a bad rap. For your budget you could get a huge arts and crafts 30s house in Highfield, right by the common. Agree with the pp that it's the best spot to live in. The local CofE primary is lovely, although Not sure on secondary school situation these days.
Southampton is definitely not "naice". It has edge, can feel a bit hairy in places, and isn't the prettiest. But it's wonderfully green, and for me strikes the right balance between interesting and community minded. Lots to do for teens. It's very diverse, and the amenities reflect that. Its also brilliantly located for London, the new forest, the coast, other parts of Hampshire. It's got a decent music scene, if that's your kind of thing.
The PP who shared the rightmove links in London is right - you could afford what you're looking for in Streatham, Tooting, or Wimbledon. For secondaries, Look for the Dunraven catchment in Streatham Wells, the Woodmansterne catchment in Streatham Common, and the Graveney catchment in Tooting (this last one is the tightest catchment of the bunch). There are more good primaries than you can shake a stick at through this partt of London. Also big green spaces (Tooting common, Streatham common, Crystal Palace park). Brixton, Clapham, Wandsworth and Battersea are all easy to get to, as are the Surrey hills or parts of Kent close to London.
The community feel here is brilliant, and very mixed- ethnically, socio economically, culturally. Lots of families from all over the world just going about their lives.
I'd also consider a look at Beckenham - officially in Kent but it's zone 4 London and very nice with great transport, big houses and good schools. Feels less "London" than the other areas mentioned, but may be worth a look.