I'm a teacher and have largely spent my career in high end London private schools. The fees here are much higher than in the rest of the country so it's a different crowd - when I was a child in prep school myself in the London suburbs 30 odd years ago, my dad was a self-employed small business owner and my mum worked part time, and they could just about scrape together the fees for me and my brother and still stay afloat and afford holidays, etc. With the cost of fees now, I can't imagine anyone in that same income category being able to manage without family help.
In the schools where I've taught, all the parents have been seriously high earners or come from moneyed backgrounds.
There's the small % of old landed money types - children of European royalty, aristocrats, politicians etc, for whom the school fees are paid for out of family trust funds. Alongside those, there's a small % of major celebrity children - think children of huge pop stars and big-name actors, for whom the fees are pocket change.
Then there's about 20% of children whose parents are self-made billionaires and millionaires - usually hedge fund managers and bankers. Again, the fees to them are pocket change.
The bulk are people with very good corporate/finance/legal sector jobs who are earning around £250-300k per year - and they pay the fees out of earnings. Sometimes it's double-earning families, but more often than not, it's a high earning dad and a stay at home mum or 'hobby job' mum (lots of interior designers who just do up their friends' houses). My school costs over £30k per year and many families have multiple kids in the school, so to be paying £60k-100k a year in fees for 2/3 children, this isn't the kind of choice you can make unless you're a seriously high earner.
There's also a small % of teacher's kids, as most top-end schools give at least 50% off the fees, which does make it manageable on a private school teacher's salary. For a lot of the teachers, that's why they work at the school.
The kids I teach don't care about designer labels. They all look like they've just rolled out of bed, TBH! They're not flashy like that at all. There are a lot of very fancy foreign holidays and so on but no one brags/talks about money or makes anyone feel lesser than if they're not in the same financial league. In my current school, we have teachers' kids who are paying hardly any fees being best mates with kids of famous pop stars. They're all equals in each others' eyes, which I think is lovely.