If you are paying 45k pa, the 20% vat would take it to 54k, i doubt that is going to worry most parents with kids at a boarding school, would half that for day pupils.
So would not lead to a flurry of parents putting their kids into the nearest comp.
With all due respect jasjas I don't know whether you have experience of private schools but in my experience that is not the case at all.
At the private school my kids attend there are a lot of parents who are definitely not rich, they live in very average houses (and sometimes flats) but to them private school is obviously a bigger priority than other things. There are also a lot of parents who have been given a discount on fees. But generally there's not a huge difference in the wealth of private school families versus the best state schools with the sought after catchments.
For those people a 20% increase would just push into being unaffordable.
We are probably wealthier than some of the people I describe above but an immediate 20% increase would make us rethink our plans. Currently only one dc is in private school but have two younger preschoolers so eventually we need to have the money to pay for fees for three kids. So a 20% increase for three would probably mean an extra £10k which probably pushes it in to being unaffordable.
So for the younger two we'd apply for the local state school and save the money for secondary or the later years of primary school. We live close enough to definitely get into an outstanding state school, so we'd just bump someone else's kid from getting a place at one of the most sought after state schools in our area.
People who will scrimp and save enough to squeeze together the fees for school will not accept their kid going to the worst state schools if they can no longer afford private. They'll move house / go to church or do whatever to make sure that doesn't happen.