An excellent school in a suburb of Hampstead called Hampstead Garden Suburb is Brooklands School where parents move from all over the place to get in the catchment area. If you go there, it would honestly be just as good as a private and many move from the private system to there. Others are Garden Suburb School, Holy Trinity, Christchurch, Fitzjohns, New End and Hampstead Parochial. These are all in NW, don't know as much about SW but these are all excellent.
We are Catholic, but not very good ones...I would pay more just to get good schools, that's what I am used to in the US and as far as transport...if my husband can get to the airports and the chunnel in 30 minutes that would be great too. I guess the thing I would enjoy the most is a nice village to do some shopping for food and perhaps daily needs. If you recommend an area that is more pricey, that's ok, I will shove all my girls in one bedroom. My girls are 4 3/4, 2 1/4 and one due in 5 days.
If you lived in Putney/Roehampton you could look at Roehampton Church School. It's CofE and it was lovely when I was there - a very, very long time ago! Quite 'prep' in its approach - girls wore boaters, boys wore caps etc. Might not be the same now though.
When you say Chunnel? Do you mean eurostar or the road equivalent ? If you want quick access to eurostar and airport then nw probably better as have St pancras (where eurostar is ) and heathrow express in easy reach (southwest ok for heathrow nut nor foe eurostar )
I would go to a place with good community primaries, as the Catholic ones don't just require you to be Catholic, for example, they require you to practice in that particular church affiliated with the school, spend a year or so before applying going to church every Sunday etc., not worth it.
in SW, I would say the best place to live for excellent state primaries is certainly Richmond/Sheen/Barnes. Barnes Primary, East Sheen and Sheen Mount primaries are all excellent. There are several more excellent community primaries in Richmond as well.
I don't think Putney/Fulham etc are great if you don't have a church affiliation.
Nowhere in SW/NW London is 30 mins from the Chunnel I'm afraid; we are very north Surrey and it's about 30 mins to heathrow/gatwick but around 90 minutes to the chunnel. We're in Epsom, and there are some great primaries (and secondaries) here.