Both my kids go to a private school, not somewhere where the fees are extortionate and whilst there are some extremely rich families there where another £3k per child will be meaningless, mostly its made up of fairly regular people. There are also quite a number of them where grandparents help out with the fees a bit, or pay them in full, professions of the parents are things like police officers, a carpenter, a plumber, a teacher, someone who works in HR etc.
I know of 6 kids across 2 year groups who won't be coming back when VAT is added, but I don't know every kid/family.
I'd say it will have quite a big effect on the numbers of kids going back to state schools, although clearly my knowledge is anecdotal. My two will stay, but my wife will move up to working 5 days a week instead of 4 and we'll scale back on things like holidays. Its worth it for me as the state alternatives around here are not nearly as good (in my opinion) and that was what pushed us into looking at private schools in the first place.