So you are open to considering both North West and South West? Most Londoners would associate West with "West but North of the River".
Would you consider Balham (SW12)? Trains to Victoria + Northern line. Depending on how far from Oxford circus you really need to go, a bus from Victoria or Vauxhall might be better than the tube, especially if you are closer to Bond street than to Soho.
Or Ealing Broadway? It's not the shortest commute from there, but you should be able to find a seat on the central line, then no changes to Oxford Circus, and you should be able to walk from Bank to London Bridge (it may be quicker than changing at Bank).
Wimbledon or Earlsfield? Wimbledon is much nicer but more expensive. You can change at Waterloo to go to London Bridge, and at Vauxhall to go to Oxford Circus.
The train lines Balham -Clapham Junction - Victoria and Wimbledon - Earlsfield - Clapham Junction - Vauxhall - Waterloo are very frequent, something like every 3-4 minutes at rush hour, so almost like the tube.
I used to live in Hammersmith and liked it a lot. Not as green and leafy as other areas, but close to High Street Kensington and Holland Park, and I really liked the Thames Path. Somewhere closer to Shepherd's bush, so you can jump on the central line without changing, would probably make your commutes easier.
NappyValleyNet is a good place to ask about SW London. Just bear in mind that it's difficult to get honest feedback because lots of people seem all too keen on convincing themselves and you that wherever they bough a property is the best place on the planet.
The people i know who lived in Putney tell me the district line is useless to get beyond Earls Court - simply too slow. E.g if you need to get to Victoria, the train, with an interchange at Clapham Junction, is probably a better option.
You haven't said what your budget is, if you have kids, if you're thinking of schools ( private, state, state but not religious etc).
Are you in a chain? First-time buyers? Would you consider renting, at least for a while, to understand if you like the area and to understand what happens to the real estate market in the meanwhile? I have been househunting myself, and my impressions are that prices are slightly coming down, and volumes are plummeting. Houses above £1m and areas that used to get the "overflow" from Kensington, like Fulham, have been suffering more. I don't have £1-2m, but if I won it at the lottery, I might consider buying south of the river, but not north. That's of course just me - everyone will have different opinions.