Hi Habanera, just wanted to let you know my DD sounds like yours.
In primary school, she found the work easy and was way ahead academically. She had a few friends, they were nice quiet girls, but was bullied very badly by some of the others. I think this was because she was bright, and a bit of an introvert. (She is a huge bookworm, and would rather read than do anything else!)
The bullying got so bad that she started to suffer from IBS. She was tested for all sorts in the hospital, but we were told that there was nothing wrong and that the only explanation was that she was suffering from stress. The only thing that she was unhappy about was school. :(
We are not in London, we are in rural south Wales. We made the decision to send her to a local private school. The academic results there are excellent, but most importantly, it is small, everyone knows everyone else, and the pastoral care is excellent. She has had friendship ups and downs like everyone else but on the whole, she loves it there. She has gone in today for her last GSCE.
I said to her the other day, that I wished I had moved her to a different primary school. She told me that she was glad she had stayed in her school even with the bullying, because that was the reason she went to her secondary school, and in her words, "I love my school, and wouldn't want to go anywhere else." That was wonderful to hear from a little girl who used to cry herself to sleep at night because she was so scared to go to school the next day!
Hopefully you will find a school that suits your DD the way we have. My DD is now very happy. She is still an introvert, just like me and DH! She has lots of friends though, and is very friendly with a group of 'geeky' boys! If you find a suitable school, I am sure she will make friends and blossom. Good luck!