I think if the high court challenge removed VAT from school fees, and if it applied retrospectively, then because you presumably have an invoice that states (for example) fees £5k VAT £1k total £6k, then yes, they would have to refund you any VAT that you had already paid. They would not be able to hold on to the VAT that you have paid, because it would sort of cease to exist.
VAT returns are usually completed quarterly and the net VAT payable is paid over to HMRC a month after the end of a VAT quarter. So it might be that they haven’t actually paid anything to HMRC yet. In which case it would be a bit simpler for them to refund you, as they should have the money sitting in a bank account somewhere.
If they had already paid HMRC then it could take a bit longer to unwind, and you might have to wait until they’d clawed money back from HMRC to be able to refund you.
But you should get it back eventually, if any decision by the high court applied retrospectively.
Schools only have to pay any VAT over to HMRC once they have invoiced it to parents. So unless you have paid loads of terms upfront, then the schools won’t have paid over VAT for future terms, that’s not how VAT works.