I know of some private schools that are delighted that the VAT rule has been introduced as the major works they were planning are now 1/6th cheaper than they'd originally budgeted for! Many very wealthy schools know their parents will/can just swallow it.
Many parents that send their children to private school either have their own companies or are employed by companies that will pay for the school fees as part of their remuneration package. Those companies then claim the VAT back. The parent is then taxed as a Benefit In Kind on the VAT inclusive cost, but 40% of the extra 20% is much more palatable for them.
I do know of someone though that has had to take a job away from home to boost his income to cover the school fees VAT. If the children had been younger they may have transferred them over to the state system, but as it is they're both in GCSE years.
Their daughter is doing GCSEs this year and has sourced a private sixth form that is a state educated grammar school but offers week boarding because even she's conscious of the impact it's having on her family.
As others have said, the true impact won't be seen yet. The new VAT rules only came in in January. Give it another academic year or two and then look at the numbers again.