Kiho, yes for very academic schools, but occasional places will come up, and indeed I have seen cases where the very selective schools will look at a genuine academic high flier if a trusted prep school head were giving a strong recommendation.
Mine were at Newton Prep, near the US Embassy, albeit a long time ago. The current head is well regarded. DS stayed until 13, and the situation then, and I don’t see why it would change, was that Yrs 7 & 8 were small and an effective split between the more academic who were heading off to 13+ schools including Eton, Westminster etc, and others including new pupils from overseas or who had failed to get a private secondary place at 11+, and those who were either headed for boarding or for other reasons wanted to delay the change. (The parents of one American girl expected a move overseas within a couple of years.)
It worked well. DS enjoyed having the responsibility of being at the top of the school, helping out with lower school sports days etc, and being part of a small, bonded cohort. Boys at 11 vary a lot. DS was quiet and studious unlike some of his peers in Yr 6 who had clearly outgrown the school and needed to move on.
Schools like Newton are very used to the adaptation. Two years in a good prep should mean that he will be more than up to speed in most systems. As a renter and you realise you are staying on and want state, you simply move home after a year so you are living practically next door to your preferred secondary, and so first in the queue for a place.
As I said, my kids are now at University, indeed DS is in the States. However we knew a lot of expats, both through school, and through my daughters sport. Some of your countrywomen started their focus on College entry at a very early stage., and would have given Manhattan tiger moms a run for their money. Most others were more relaxed. Do PM me if you have any London questions.