Unfortunately no one knows the answer, including the Labour Party. No one knows how the proposed changes will be implemented.
First, a few years ago, the Labour Party voted to abolish private schools altogether. They realised that was pretty much impossible so scrapped that idea.
The next idea was to scrap charitable status of private schools. It took them a few years to work out that half of private schools aren’t charities anyway, and changing charitable status would have no impact on that anyway. So they scrapped that policy.
The latest idea is to add VAT onto fees. This is a relatively recent plan (in the last few months) so hasn’t been fully thought through (just like the previous proposals).
The problem with that is that the current VAT exemption is entrenched in quite complicated legislation. Changing that without having unintended consequences will be extremely difficult. For example, at the moment, university tuition fees are exempt from VAT by exactly the same part of the legislation as private schools are.
It will be difficult to say that after school clubs and wrap around care provided by private schools will be subject to VAT without inadvertently catching state provision within the new rules.
The proposals are estimated to raise a tiny amount of money, if any. Changing the VAT legislation and risking catching other areas inadvertently seems like a ridiculous idea.