Chartered accountant here.
Of the total amount that the school bills, 1/6 will be payable to HMRC as VAT.
So a school might have billed £10,000 per annum pre VAT imposition. It could bill £12,000 after the new budget, keep £10,000 and pass £2,000 to HMRC. Or, it could bill £10,800, keep £9,000, and pass £1,800 on to HMRC. Some schools have effectively planned for this by charging above inflation rises over the past few years, so that they can absorb part or all of the VAT in due course.
VAT is charged to schools on some of there costs (not salaries which are by far the biggest). But certainly things like maintenance will carry a VAT cost which the school will be able to reclaim against the VAT it has charged on its bills. It won't be massive, but will reduce overall costs by something in the order of 1-5%-ish.
I think it's probable that the increase in the proportion of pupils that the state educates will cost more than the amount of revenue raised by the VAT. Hopefully, the attendant influx of sharp-elbowed parents into the state sector will help to improve standards there.