there is a sailing school called Hamilton Sailing. they conduct classes somewhere near to The Crown pub as they come in afterwards for drinks sometimes.
i think they do beginners right up to skippers classes
and they take the classes sailing on weekends.
dont have a contact other than the instructors name is Mark.
I'm with the sailing club at Shadwell Basin. There are courses run through the centre from about April to October, have a look at http://www.shadwell-basin.co.uk/ for more information. The sailing club meets on Tuesday evenings during the same months, sailing open to all, no experience necessary, more info at http://www.shadwellsailing.com/
I did the RYA Level 1 and Level 2 courses at the Westminster Boating Base http://www.westminsterboatingbase.co.uk/ a couple of years ago; great fun and amazingly good value (works out to about £10/hr).
You learn to sail along the river between Vauxhall and Battersea - being allowed to pootle about on such a busy stretch of the river is a treat unto itself.
I did the adults' course (it's primarily a charity aimed at getting kids out on the water) and met some great people too. Highly recommend it! I learned enough to be able to hire a small dinghy when on holiday in the Med (which was a LOT easier than the Thames ;) )
PS - I enjoyed it all so much I also did the powerboat training course a month later! Also fun, but you spend more time on the water if doing the sailing course.
I did RYA 2 at WBB ALSO, and had a similarly good experience.
Being objective, the only drawback is that learning on the Thames involves constantly fighting the current - which is great if you plan to sail in tidal waters in the future, but possibly not so useful if you plan on sailing on a lake.
Welsh Harp (well out of SE1) is a friendly lake-based place to learn. And they have a nice clubhouse.