The whole state system right now relies on some parents paying twice for education - once through income taxes and once through private school fees.
Right now the state has X amount of funding from taxation to spend on Y number of pupils.
If private schools are abolished, as some would advocate, the pool of funding stays at X* but the number of pupils it has to educate goes from Y to Y + 600,000. Which means even less money can be spent per pupil than it is today, OR money has to be moved from other things, like the NHS, just to keep spending per pupil where it is now.
Buying land and building hundreds of new schools is very expensive so let’s assume that the state saves a bit of money by compulsorily purchasing private school buildings below market value and rebadging them as state schools. But what happens then is that house prices rise even higher in what are already expensive places to live close to these schools. Essentially the same kids carry on going to the same school, except for free... while kids in poorer areas carry on going to the same schools as they were before, except there’s less money to spend on them. (Because who’s going to be paying the salaries and pensions of all the former private school teachers, and maintaining all those old buildings?)
The solution usually advocated at this point is “OK - let’s make catchment areas huge and have lotteries for places for every school so it doesn’t matter where you live”. (Champagne socialists who live in multi million pound homes next door to Holland Park Comprehensive or the West London Free School tend to get a bit less keen about abolishing private schools at this point...)
We could do this, but there would be consequences for everyone. Less walking to school and more carbon emissions, because huge numbers of kids from the age of 5 upwards would have to be bussed miles across cities and counties. (Who buys and runs all the buses?) Kids might well end up having no local school friends at all. Parents would effectively lose all choice over where their kids went, even a choice between state schools, as it would entirely be done by lottery.
Personally I think there are better ways to improve substandard state education and address inequality than wasting millions abolishing private schools.
*Contrary to popular belief, tax relief on fees for private schools that do not exist anymore does not raise any money for the Exchequer.