I'll describe the grammar school system here in Kent, which is adjacent to London. Over here they call the selective government schools "Grammar Schools". Grammar schools are free, and the top grammars send a comparable number of kids to Oxbridge as the top independent (aka Private for internaional readers, Public for British readers) schools. For reference the most expensive school in the area is also one of the most expensive schools in the country, at about £33K a year for day and £45K for boarding.
The year the kids turn 11, (year 6), they start the year with a test. It looks a lot like an IQ test, complete with silly questions like "what does this shape look like from this direction". There's also a bunch of standard stuff like arithmetic and vocabulary. It is a single sitting on a single day, so try not to be ill.
The thing is, and I didn't know this until after we got the results, almost every kid who gets in had a tutor for the test. My kid was getting fewer and fewer playdates after school in years 4 and 5, and I suspect this is why. Nobody admits to hiring a tutor for some reason, but everyone does it.
The kid then gets in anyway, and what do you find?
Most primary school kids were not privately educated, but most of his classmates came from private schools. He went to one, and this school has the highest representation in the new intake. I met the other parents, and it's just a selection of other private schools in the area. Maybe one or two from local state schools. I got my kid to ask, and sure enough everyone had a tutor.
In terms of ethnicity, it's pretty mixed. There's more Chinese/Asian kids there than I suspect there are in the local population. There's more Indian/Asian (Asian in the UK means India/Pakistan/Bangladesh/Sri Lanka) kids there than I see in everyday life. There's a few black kids as well, and a lot of mixed kids. The majority is white, as you'd expect.
So if you care about race, you'd probably say it's nicely mixed. If you care about social mobility, I think you might be disappointed.