This applies to non selective state schools and probably isn't helpful if you want to find him a place in an independent or selective school,
School places will have been applied for last autumn and families will have heard in March. Some will have appealed etc. Many schools aren't oversubscribed but those schools that are will be the ones with a reputation for really good GCSE results, particularly if they also have an outstanding OFSTED rating. In such cases, it doesn't matter if you find a house on the doorstep of such a school, because if they're really oversubscribed, being close enough to have a good chance of a place if you'd applied last year isn't really relevant.
There may be plenty of places at schools which still offer a lot of their students excellent teaching, where some students will learn and do fantastically well, but that are in in less affluent areas and have a more varied intake.
Education is quite unequal in some boroughs, like mine, and very high proportions of amazing GCSE results may be partly due to the intake to start with, and to families who pay for additional support etc.
You probably need to narrow down the areas that you will consider moving to on your other criteria and then find out about all the schools. Try to find out about more than raw results. Do they do that by restricting entry for exams/signing up for options/offrolling/squeezing out pupils who are unlikely to get high grades. If your son is into sport/music or anything else beyond an academic curriculum, what's on offer there?
Particularly as you're moving across London, might it be better that he can go to a school which could teach him well even if it's not the most competitive and oversubscribed school in the area, and where most other kids will live fairly locally, or where he can pursue/develop extra curricular stuff if that's important.?
Good luck.