Hi PJP, congrats on your acceptance to your masters program.
I'm currently living in SW London -- Vauxhall. My husband and I share a very very very tiny (no seriously, I'm not kidding, people don't believe me; it's
minute) studio flat. Let's put it this way, we can only stand upright in the middle of the flat because it's an attic. We can fit a bed, a table and a desk along with the kitchen fittings (including the world's smallest oven that doesn't fit any of my cookie sheets from the States). We must walk up four flights of stairs to get to said tiny tiny flat. We can't invite people over because we can't fit more than one person, and everyone bangs their heads anyway.
I'm saying this all for a reason. We pay 650 a month, and I'm panicked that we'll have a rent increase soon. Yes, Vauxhall is still zone 1 (border of zone 1/2 actually) and I know we are lucky. I do recommend my neighborhood, but I am not confident you'll find anything much in your price range (and sorry, we're not moving soon
) You could luck out of course. I think you may have to look outside of zone 1, but, if you can find something on a good bus line the travel costs won't be exorbitent, or too time-restrictive. For example, looking in zone 2 will up your potential housing immensely.
I don't know much about other neighborhoods, but I'm happy to expound on the virtues (and dis-virtues) of Lambeth Council.
Where you look for a place is really going to be based on where you and your wife need to get to every day. If, for example, either of you is working/studying near to a train station, you may consider going further out into the suburbs where you can get something lovely for less, and the trains are (despite my complaining) fairly useable.
HTH,
Tricia