The OP asked how it would work in reality, and that's what I'm wondering too.
So, what would happen that if in September 2020 all private schools were abolished?
If I look at secondary schools near us I would imagine that all the places in the main entry years Y7 for girls and Y7 and Y9 are already filled, and at least some of the intake for 2021 has already been sorted. So there would be limited number of places available for pupils not already in the system. And there would be tons of siblings going forward. If Labour loses after the first term, there should not be a huge impact on the pupil profile.
Is the Labour government proposing to scrapping all academic selection as well? i.e. grammar schools? Round here most of the good private schools are academically selective, some exceptionally so. I am not for one minute suggesting that there are not bright kids in the state sector, but if academic selection is not ruled out, you will just have a handful of pupils coming through who would have got into grammars anyway. This policy is not going to help the kids who are struggling in the comps with low level disruption.
Also. are they going to put a cap on the price of uniform/ extras etc? Or limit teaching styles? Or school policies? You can bet your bottom dollar that the average working class family living near Harrow will not want to fork out for a blazer if the school put the price up to £150 and say that uniform is compulsory.
If I was the chair of governors at a prestigious private school I would just raise the academic selection criteria, or have specific entry requirements.
We can't get into our amazing local comp because we are the wrong religion, so we're going to pay for our DCs to go to a great private school. However, if the private school is destroyed, we'll be sending our quite bright (but not brilliant) children to a good local sixth form.
Being brutally honest, they will probably get better GCSE results than a lot of the kids from the more deprived parts of town. So we will take their sixth form places. This is a dick move on our part, and I am thoroughly ashamed of myself for even thinking about it, but that is the reality.
Also, do Labour seriously think that the culture of private schools will change overnight? Some of these schools have been round for centuries, they are not going to roll over and take this. They have large reserves of money and loads of connections. They will fight it through the courts, and if needs be, sit it out for 4 years until the next election.
Oh, and I have been a Labour voter in the past, but they've lost me for the foreseeable future.