Private schools have never had a VAT exemption. Before Labour introduced this policy the schools paid VAT on all purchases and were unable to reclaim it, unlike most businesses who collect VAT from their customers on behalf of HMRC. Now that the schools will be collecting VAT from their customers (fee-paying parents), they will be able to reclaim VAT. So, next time Eton build a swimming pool they will be able to reclaim any VAT charged on materials and construction. State schools have always been VAT exempt so do not pay on their own purchases.
It is the purchase of educational services that was previously exempt. Ie the parents were exempt and now are not. Although this tweak applies only for private schooling, not any other kind of education.
I wish people understood these basic points about how things work!