I think it's far too soon to tell what the overall impact of this will be. Most parents will understandly be very reluctant to move their children out of private school mid-key stage and they will be borrowing from their own parents, remortgaging and taking out loans to get them through to an appropriate time to move their children. This will be especially true if the only state school places available to them are the ones nobody wants, and that's quite likely if they are trying to move mid-key stage.
You'll see a big difference when it comes to new registrations at reception level, then year 6 and then sixth form though. That's when children who might have gone to private school now won't be going, and some of those currently in private school will be pulled out and moved into state.
For the ones who need to move school, their parents may be looking at moving house or even moving to a completely different part of the country to get an acceptable state school place, or they might be working on a plan for one paretn to give up work and home educate.
What will be happening is that people who would have liked private school but are now questioning the cost and the benefits will be sinking all their efforts and resources into securing housing in the catchment of the best state schools and spending their money on tutoring for the eleven plus in grammar areas.
The stupid thing about this whole thing is that if you are fundamentally opposed to private education on the grounds that it benefits the privileged (it does) and disadvantages lower income kids (it doesn't) then if you abolished it tomorrow that really would be a problem for state schools. Because those parents already pay a contribution to state education through the general taxation system but are currently not using the places their children are entitled to. If every child left private education then state schools would need to absorb them all but not a penny of extra money would be coming their way under the current arrangement.
Yet this private school tax raises a relatively small amount of money and some schools have already found perfectly legal accounting loopholes that mean they can claim back more from HMRC than the government stands to collect in extra taxes from the parents at that school.