If your dd has any friends, you could always try and book a private swimming teacher thru your leisure centre and share the cost for a Sat am slot. I know that most Wandsworth Leisure centres offer various teachers to do classes that way, although I don't know what age they start at. Ds does them at the Latchmere in Battersea, teachers are great but he started at 4 which seemed like a good age for him (he wouldn't have listened to the teacher before that). Otherwise, have you tried places like Aquatots?
Re the friends thing, yes she does but I think it's something dh wants to do just with dd rather than other people we know so he would just end up chatting to them and not spending his quality time with her (pass the bucket at that phrase!!)
Whereabouts in SW London? I took ds for Saturday morning swimming lessons in Clapham pool, Clapham Manor Street. It's pretty grotty though, but the prices are good and the teacher excellent. If you do go, don't whatever you do use the children's changing room as it's chaotic. Use the adult one.
DD takes lessons through an organisation called Swimming Nature. Their literature says they teach at the Chelsea Sports Centre, Jubilee Sports Centre in W10, Kensington Sports Centre in W11, Cannons Health Club in Norbury SW16, and other places that wouldn't be convenient for you. You can contact them on 08700 949597.
We take during the school term (after school), but they also do half term and summer fast track courses. Not sure about weekends. We've been very happy with them.
I also know one mum who takes her dd for lessons at one of the the David Lloyd Health Clubs - I think it's the one in Kensington. They are not members of the club, but somehow are allowed to come in for the lessons only. So, something like that may be an option too....though again, I have no idea about weekends.