It’s hard to say really OP, without knowing your housing budget.
If you don’t want full on suburbia, but want a more rural feel and more space, could I suggest somewhere the other side of Richmond - eg. Petersham or Hampton or anywhere along that meander of the river? It’s fantastic for families. Very green. Obviously it’s not cheap by national standards, but your money will buy you more space over that way than say, in Barnes or Chelsea.
There is a boys school in Hampton (called Hampton, funnily enough) and they achieve about 95% A-A* at GCSE. St. Paul’s used to be the aspirational school that everyone knows about, but these days, not so much, as many schools achieve very similar results. There is also Kings Wimbledon, for instance.
We have friends who live in Edinburgh with kids at Fettes. It looks very grand, but it’s very long days (obviously due to the boarding style) and the results are what would be considered low-average in terms London of London independents. They are very frustrated with it actually and moving.
But the London schools are very competitive and don’t suit everyone, obviously. At 4 months, you have no idea if your DS will be academic or not. Or have a SLD such as dyslexia.
The problem in the independent sector in London is that the top maybe 2-5% will get into the schools that regularly feature in the top 10 in the league tables. But then, the schools that should be catering for the children of more average to high ability, are snapping at the heels of the top schools - raising their entrance requirements to snap up the ones who don’t quite get in to the very top schools. And so it goes on. So the kids have to sit about 5 schools, but even the “back up options” are schools that get about 75% A-A* at GCSE (which is the same as a school that would be considered aspirational elsewhere eg Fettes).
The other thing I have seen with friends that have moved out Berkshire way for more space (eg Henley and places like that) - well, that area is boarding school central and so, not surprisingly, all the preps are geared to 13+ and feed the boarding schools west of London - eg. Wellington, Eton etc etc. These are less competitive than the London Day Schools though, simply because the market for boarding is smaller.
Kent has Tonbridge School which is easier to get into because it’s boarding (but very good results).
But there are few “softer” schools in terms of entry requirements in SW London into Surrey area, such as Reeds or Epsom College, for example. The results will be similar to Fettes (did you say you were planning on Fettes - sorry if not)!
Personally, I would rather have a slightly smaller house in an area where there’s more going on - but everyone is different!