OK, so I've been reduced to using this for now (I have no idea who londondpropertywatch.com are?) but they're drawing down a sample of contemporary asking prices (an asking price needn't be achieved of course) on 5965 properties across 4 different property types defined by number of bedrooms, across 69 postcodes in the capital
The average ask for a one bedroom unit is £364.73 a week, for an annual property overhead of £18,966 if multiplied by 52 weeks
http://www.londonpropertywatch.co.uk/average_rental_prices.html
Incidentally Clive, the average asking price for a 3 bedroomed property in Twickenham was £586 a week and £1132 for a bigger property. This is obviously significantly larger than your qualitative evidence of "about £1600 a month now and I think it was £1200 about ten years ago."
Can we deduce that living next door to you is possibly having a local impact on the market?
I should say, I'm reasonably sceptical about the source, as they do sound a little bit like a pressure group who possess an agenda (albeit some pressure groups can of course be perfectly credible and are routinely quoted - the government is fond of quoting Human Rights Watch despite their very chequred history of impartiality).
I did try Zoopla but couldn't find any research, albeit they were delighted to act as a letting agent for me!
It's a shame the ONS aren't prepared to attach their own spreadsheet to the index they've built