One month in London

Definately register with couchsurfing...you've enough time to get some references by hosting yourself ( even if you just meet for coffees etc). You can then stay with locals all over the country and see the real places! We did it in Iran and it was the best thing we've ever done, now hosting in London, great way to meet interesting people!

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[ Edit: Sorry, no promos please. ]

I think it's a lovely idea to spend more time on one place and really exploring it rather than just passing through. If you're planning to travel all around the UK perhaps you should consider staying somewhere other than London, as, like everyone else has said, it probably is one of the most expensive places to stay. Perhaps plan a stay in one of the other cities you want to visit? Or even stay in a suburban area, just outside of a city? Aslong as there's a train station you'll still be able to get around very easily. For cities to visit, I'd recommend London (obviously), I love Edinburgh, Bath is very pretty and Bristol's nice too.