Schools are unable to say what the additional fees are because as yet (I work in school finance) we don't know what the policy is regarding the VAT addition.
The headline is that providing the service of education will attract VAT at 20% - but a school bill is made up of education and other components, for example catering and boarding, currently these are VAT exempt if provided in the course of education, no one knows if this will change or not.
If certain elements of the bill remain VAT free, then the overarching percentage increase to the total fee will be less than 20%.
Once the full details re known school bursars can look at weather, but gentle manipulation (ie slight reduction in education, slight increase in VAT free services) that the overall effect can be minimised.
My guess, and others I have spoken too in the sector is that the devil will be in the detail, and HMRC will have thought of most workarounds and plugged them!
The schools could reduce their fees and therefore reduce the VAT payable, but cannot reduce the VAT itself.
As an amusing aside, I was talking to the SBM of our local Pupil Referral Unit, which is funded by the LA on a per pupil fee basis, built by the LA in the 60's but a charitable trust registered as an independent school - from jan, they will have to charge VAT to the LA, which raised a wry smile😬
Another point to remember is many private schools, like mine, are not VAT registered, we have just registered in readiness - this has an effect of course the other way, that we can now claim VAT back on our purchases, hitherto never done - so although we have to add one end, like other VAT registered businesses we can claim the other end too, which will be a significant saving.