I would also like my 2 yr old to start swimming lessons-she did do some baby swimming as a tiny baby but we stopped because she wasn't enjoying it after a couple of terms.
Anyway, I was looking into it, and London Baby Swim (who we went to originally) are opening a site in Wandsworth (at/near Southside I think) in April.
Beginner Toddler lessons seem to be on Sundays which isn't an option for us, but might be for you? Online it says there is one space left:
http://www.londonbabyswim.co.uk/bookingpage
I have to say, they were lovely when we went when my daughter was a baby, but Osterley is just too far (where their original site is).
Hope that helps