Anyone who thinks that adding VAT to school fees (if indeed this is the plan rather than scrapping charitable status which is particularly complicated) is going to benefit the state system is going to be disappointed.
What it will do, if it happens (and I suspect there will be years of legal wrangling first) is drive many parents into the state system. Which is already underfunded and too full. House prices near good schools will increase and yet more people will be forced into weaker schools.
The 20% VAT doesn't cover the cost of the state education, the state schools wild potentially have more pupils and less money.
We privately educate, I was state educated and it was truly awful.
If they charge us another 20% we'd pay it. The weaker and smaller private schools would close.
So you'd be left with: an even more elite private school system for the very wealthy, without any charitable benefits, and an even more divided state school system.
And if you shut private schools (which would never happen) we would 'home school' with friends, paying for tutors and teachers who perhaps used to teach at private schools. And we'd pay for the sports and music lessons which you cannot get at state school. And it wouldn't be a 'private school' but it would be a private home school set up.