Are you planning on working as well? Otherwise you will find living two people living in London on £50k very restrictive, especially spending £300/week on rent.

Between Knightsbridge and Hammersmith you will find there is a huge difference in price. In Knightsbridge I'd expect to at least double your budget for a 2 bedroom apartment. In Hammersmith it's doable but would still probably be closer to £350-400/week for something half decent.