If you earn towards the end of that range you should be able to chose pretty much anywhere in town( at around £1200 per month rental), of course in Notting Hill, Kensignton and Chelsea and other posh places you would not get more than a small studio flat (i.e. a glorified bedroom where everything except the bathroom is in the same room). in other areas (like Greenwich, Islington and even around Waterloo) you could manage to find a one bedroom flat.

If your aim is to save money just flat share, then you could pay around half of that to share a 2 bed flat somewhere middle of the way (Greenwich and the like).

If you are earning more towards the low end of the range, then most likely you need to look for flat sharing somewhere East of town or South of the river, but in any case further away from town centre.

Either way, another option is to live out of town, where housing is cheaper. In Woking for example, half an hour away from London, you can get a 3 bed house with garden for what you would pay to get a 2 bed flat in Greenwich (fact, I own both), unfortunately you would pay a fortune for train fares (around £3000 or more a year) for a service that leaves lots to be desired (I spent most of my time in London to avoid the trains as much as possible).

I hope this helps to inform your decision.