My daughter started there this year for sixth form. Presumably yours would stay on, the vast majority did, partially due to the pandemic making looking at other schools difficult. So some of the following will not be as relevant to you but gives a flavour I hope.
We live in Wimbledon, are comfortable but not well off - I am a widow. I'd say we are less well off than the majority, many of whom live within walking distance of the school, but I don't think that has had any affect on my daughter - she's not a fomo person anyway.
They let only 15 or so girls in for sixth form so it was a high entry standard, and I have to say i have been surprised at the academic pressure, as I thought FH was less academic than some (say PHS for example, which required 4 to 5 predicted 9s and the rest 8s for entry, even though that is not their published level. My daughter applied snd was not offered a place). She got 8s and 9s on her GCSEs so is used to working hard but says she has an even higher workload, particularly in History.
It is true that there are few or no cliques! Everyone is nice, though my daughter has yet to connect with more than a couple girls, but her classes will be a lot smaller and fewer than Y7, also she is not particularly social. I think lack of cliques also cuts out much bullying.
The teachers are good and she likes them, obviously some more than others. They have an interesting guest speaker program which she very much enjoys.
There are tons of clubs. They are not known for sport but there are opportunities to participate. There are a number of school functions for parents.
In sixth form they are required to volunteer (my daughter helps out with music therapy for autistic children), which is good. They also must pick extra classes outside of their subjects: my dd did philosophy for one term, which has led us to meat free Mondays, and will do creative writing this term (cooking and diy are other classes that can take and more). It's an interesting alternative to their three subjects and as no exams they can be enjoyed rather than adding to their workload.
There is a good library and plenty of study places. Food is very good!
All in all my daughter is happy we moved there, now that she had figured out the school routine. She feels safe and comfortable. She has a long journey walking plus the tube, but walks back at night to the tube with a friend (who goes the opposite way) and has learned to deal with the odd nutter and drunk football supporter!
I think they do quite well integrating the new girls and they are of course very pro women, though like most all girl schools need to remember that not all girls want or should be aiming to be captains of industry! But their expectations are high, and I'm not sure how they are about alternatives to university.
So to sum up: a welcoming school, mixed yet higher than average (even for private) on the socioeconomic scale; quite academic, friendly pupils, plenty of opportunities to get involved in different activities, quite full on sixth form (EPQ, volunteering, speaker program, extra classes). Head of years very approachable and do check in with the girls, and had lunch with the new ones. Form tutors also keen to get to know the girls on an individual basis. I'd say pastoral care is good, though it does rely on the girls seeking help if needed (which is the case everywhere).