Hi - we had exactly the same issue last year. My daughter was in year 1 at a local state primary and as you say wasn't quite what was expected. However, we called around a few local private schools in the area in the December and she is now firmly settled at a local private school in year 2 and absolutely loving it - it was the best decision we made. and there were 5 other kids in her year who also moved from local state primary schools so there were a few places.

Do PM me if you want more info. purposely not mentioning schools on here as don't want to get into a debate!!!

How old are the younger siblings - when will they need to start reception?

So sorry to hear your daughter is not having a good time.
I think generally it's quite difficult to move - as it means someone will have to leave the private school for a place to become available.
However I know of people who manage to do it, for example in Y1 to Newton prep (from another private school), in Y4 to Dolphin (from a state school).
I'd say register your interest with the schools you like, visit the schools (so you meet the registrar and the headmistress), make sure they know you are keen and keep calling them.
I think most parents wait for the end of the school year if they move out of London, so I'd expect the registrars to have more info in the 3rd term.

You should try to send the younger siblings to your preferred private school from reception, otherwise it is harder to get in.

good luck!

Looking for some advice...my daughter is currently in year one at a local state primary school. It's not panning out quite as we'd hoped (for a few reasons both personal and to do with the school) and we are looking at moving her to a private primary school from year two (or sooner). I've only contacted a couple of schools in the Wandsworth area so far...but it seemed to be that we've missed the boat. Is this really the case?? She has two younger siblings (a boy and a girl) so we'd be looking to send them to the school too from reception. Any words of advice? It's really hard seeing her unhappy and unable to fix it.