I don't know if you've spent time in London before Marc, but be aware that it's a huge place.
No ... really huge.
Some good suggestions given above, but many of them involve far more travel than you might think. People think Barnes is close to Twickenham because it's only a few miles away and there's an occasional bus between them.
Twickenham itself contains quite a bit of music action - see Lemonrock for a quick & easy source.
Extend your search to the neighbouring bits of West London (Kingston, Isleworth, Hampton) and there are some very good jams (Red Lion in Isleworth gets my vote) and some very good music pubs.
If you're going to be in London on 24th September, come along to the South East BassBash and meet some of us ...