I think there is a real mix.
Some have budgeted to pay school fees but not uni costs, beyond topping up maintenance.
But lots of privately educated kids have it all paid for them - but as has been said, people don’t talk about it - that’s parents and kids, because people are aware they are a minority and it’s sensitive.
I think school fees are often contributed to by grandparents and the same goes for uni costs too. With people more aware of inheritance tax, giving grandchildren £10k a year for uni fees can be a good way to avoid tax. Grandparents with surplus income can fund fees by monthly gifting £750 per month, with no 7 year rule, if funding out of surplus income rather than capital. There are a good number of wealthy grandparents who have generous pensions or interest or dividend income who can afford to do this and are also avoiding later tax.
Another category loan their kids the fees on an interest only basis with flexible repayment terms.
With all of it, thinking ahead creates more options for lots of families. Deciding when kids are in 6th form and if you have limited savings/capital and are still paying mortgage will mean most can’t fund the fees on a monthly basis from salary. But lots of families with similar finances might have put away a couple of hundred wuid per month for 10 years, and with a bit of growth, would have the £27k for tuition fees.
Of course many could never afford to cover it, regardless of careful planning ahead, but many families have enough money to make some choices - some will go for more holidays or bigger houses or newer cars etc. Others who can’t afford private school fees might decide that across 10 years of more restrained holidays etc, they could cover 3 years of tuition fees or maintenance or both. Others will select to save for a house deposit. Lots of different options and always more by thinking ahead, for most.