Moving to the UK during Year 10, what private schools in London can I apply to?

(I thought that this subreddit would be more fitting than the GCSE one.)
It's not decided yet, but according to my parents there is a high chance that I will be moving to London in April to November (probably around April). I currently live in Japan, and go to an IB school here (classes and everything are in Japanese) but I've only lived here for around 3 years, and before that I lived in the UK when I was 5-8 years old and the US when I was 8-12 years old. I am fluent in English (first language), however my grades currently are not high at all.
This may sound greedy, but I want to get into a top private school in London, and whilst researching I found that you have to contact the school directly and send an application, and if it gets accepted the school will send a reference number. I don't really know what that is, so I'm a bit confused. What will the schools ask for?
In terms of certifications, I have the ABRSM Grade 8 and ARSM. Also C2 Proficiency. I also took the English GCSE early (this year), and got a 6 despite missing the first half due to technical issues. I think that if I study the material in the UK until now, I won't have any issues if there is an entrance test (I think).
So that's my academic position (?) right now, and I did go to FHS when I lived in London.
I don't know if it is even possible to enter during the year (I really hope it is), but I would like to know which private schools in London might be possible to get into during the year, and from abroad. Possibly schools like Godolphin, St Paul's Girls school, City of London School For Girls, and such.