It may happen that under a Labour government that the VAT exemption is removed, but this is unlikely to bring in anything close to the £1.6bn being bruited about, because:
Once, the VAT exemption is removed, private schools will reclaim the VAT paid on outgoings like any other business, so the 20% figure is likely to be reduced to about 5% of receipts.
Two, the range of pupils leaving when their parents can't find the additional sums is estimated at between 5% (by the Labour Party) and 25% (the Independent Schools Council). If the latter is correct then any receipts will be negligible as the state will have those pupils added to state school rolls.
If forced into VAT registration, then a large proportion of any VAT paid for investment in buildings and facilities over a decade or so would be reclaimed by the schools, as they would be permitted to do. Quite a windfall for some schools.
And finally, as it is possible to pre-pay school fees ahead of the introduction of VAT, the exceptionally well-to-do would fund fees upfront -- saving themselves 20% VAT minus the interest paid (if they borrowed the money). If they had it tucked away in the school fee fund, they would forfeit only the interest income.
In other words, it's a bit of a red herring.