The Labour manifesto and it's accompanying funding document 'Funding Real Change' has now been published.
Contrary to my previous answer, and the simple policy details in the manifesto, it seems that the Labour party do intend to charge VAT on private school fees.
Page 2; Additional revenue raising measures
Reverse cuts to inheritance tax and Bank Levy, impose VAT on private school fees, scrap Married Persons Allowance, introduce a second homes tax 5.2
Page 13; Additional places required as a result of VAT on private school fees
This assumes that the full cost of the VAT is added to school fees with no reduction in pre-tax fees and that all of the pupils affected move into the English state system rather than being schooled elsewhere. If either of these were untrue, the cost would be lower
VAT on private school fees
According to the Oxford Economics report ‘The Impact of Independent Schools on the UK
Economy’, Independent Schools Council schools received £7.83bn for “core school operations” in 2017. Applying the same assumptions on elasticity of demand as in our funding for new state school places (Section 2) suggests a reduction in tax base to around £7.4bn and therefore a potential tax yield of just under £1.5bn.
As VAT-payers, private schools would be able to reclaim VAT on VAT-able expenses, but most outgoings relate to staff. Figure 11 of the Oxford Economics report suggests £189m of taxes on school purchases were paid, which is an upper bound on the VAT that could be
reclaimed. Deducting that and adjusting upwards for the fact that only 85% of independent schools are ISC schools gives a potential yield just over £1.5bn for 2017 which is adjusted for school fee inflation in 2018 and 2019 and inflation forecasts going forwards
There is no indication of how this policy interacts with EU law. So while I'm aware I'm not really answering the question, it seemed important to confirm that this is a genuine policy position.