We all want to I still that and so do most schools!
Don’t discount state schools, London has many excellent, high achieving ones, with excellent values. If you are moving why are you not moving to a great catchment area to hedge your bets?
One thing I will say is over the last 25 years I’ve seen the journey of quite a few families (neighbours and relatives) that chose feeder private schools at either 4+ or 7+.
The latest two families say, with hindsight, their DCs would have benefited from a different school for either prep or secondary (or both). Both also had to pay for outside clubs and lessons and wish they hadn’t gone private (at least for the whole journey), as they don’t feel DCs have done any better than friends with DCs at good state schools.
But this was c.15 years ago when immigration/birth rate was very high, hence their relief at getting a place at a highly regarded feeder schools and sticking with that school to the bitter end (and avoiding the 11+). Now, as PP said, it’s much easier to get a school place and movement is high as people relocate or simply drop out of private due to cost pressures. So you really can keep your options open.
Start local and you can move when older if you so wish. Your DC has amazing parents that want the very best for them and that is what really matters, not the perceived marginal gain of one highly regarded private school over another.