Actually you'd be surprised - public transport in London is actually really good
(I have had this debate with a few Londoners who disagree with me, but it's only when you leave London do you see what bad public transport looks like!)
The last time I was down to a Twickenham I managed to get by with public transport really easily - the tfl website makes it really easy and they update it with all the transport works in progress on a daily basis. (www.tfl.gov.uk)
That also saves you the problem of finding not only someone to put you up, but also people to provide you with transport!
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