The good news is that there are very few native speakers in London :).

I'm still nowhere, but three things helped me a lot at the very beginning:

1. BBC iPlayer http://bbc.co.uk/iplayer - it may not help a lot, but is good fun :).

2. I put my CV online - the very next minute recruiters started calling. They're 99% completely useless (even more if you're not looking for a job, haha), but you can have the same conversation over and over again :).

3. Chatting in the pub. The circumstances are the absolute worst, so if you manage to get across anything in the pub, you will be much better off under better conditions.