Non selective state school for girls in London?

PS As far as I know, all state schools in the area are non-selective when it comes to in-year admissions like yours. Those which use tests, be it for partially selective places or for banding, do so only for Y7 admissions, not for in-year admissions, like yours would be. But do check the admission policies of the schools for details.

Wandsworth's official stats are: https://www.wandsworth.gov.uk/schools-and-admissions/admissions/school-admissions-statistics/

These are the admission results as of March and as of August. Other schools / councils may publish only those as of March.
But I am not sure any school publishes what the waiting list is like for in-year admissions in Y8 - you will probably have to contact each school for that.

OP, sorry to hear about your situation.
In-year admission are tough. There is much less movement from secondary schools than from primary ones.

If you are in Putney you probably already know that Wandsworth has a mix of very oversubscribed state schools (most of which partially selective in one way or another) and very undersubscribed state schools. But even the schools which are undersubscribed on national offer day may not have place at Y8 if they form all the classes by accepting the kids which didn't get a place anywhere else.

The first thing you need to do is contact local schools and ask them about the waiting lists. Your choice may, realistically, be very limited.
Do check the criteria for in-year admissions.
If a school has, say, 30 kids ahead of you in the waiting list for criteria you cannot meet (eg special needs, or faith criteria you don't meet, etc), then spending a fortune to move opposite the school may be a waste of money.
Distance from the school tends to be one of the key criteria, but it's typically after special needs, siblings and a few others - do check the details.

You mentioned you are open to moving. It's good that you are renting, because selling a house may take a while, and if you rent near a school while still owning a house elsewhere, the council might see it as potential fraud.

If you want to genuinely move near a school to be as high as possible in the waiting list, do your research. You can find a house near St Cecilia in Southfields, but good luck finding one near Graveney.

Appreciating that choices may be limited, what is, ideally, important for you?

• Where do you stand on strict vs nurturing? Eg Ashcroft is a Marmite school with a reputation for being very strict; St Cecilia for being more nurturing, but results are not as good
• do you want a girls only or are you open to coed?
• How important is it if they use sets, when and for how many subjects?

The schools mentioned are church schools so unless you the OP has the church paperwork , will be tricky ….

The tricky situation unfortunately, is finding an outstanding school that still has places available. Most will be full or have waiting lists I suspect. The best place to start would be finding out which schools have places available.

People say good things about Waldegrave in Twickenham and the Sacred Heart in Hammersmith.

If a reduction in fees would work then do also talk to the school - I'd imagine they would want to help you allow her to at least finish the year if they can

Have you looked at Saint Cecilia’s or Lady Margaret’s? Both excellent schools for different reasons. We have been very happy at Saint C’s in Southfields - nurturing, strong in maths and music , great careers lead. Lady M’s is probably a bit more academic