The amount of muddled thinking on this subject is quite extraordinary.
The VAT exempt status of private schools is NOTHING to do with a tax break for private schools, since they already pay VAT on VATable supplies but cannot reclaim it since they are not VAT registered themselves. It is also unrelated to charitable status (which the Labour party have already decided not to remove because ultimately it would not be possible).
The tax won't be levied on private schools as such, but on the parents who pay the fees. As those who use after school clubs have noted, if something that already stretches your budget to the max goes up, it could cause you a problem.
Should private school fees become subject to VAT, private schools will need to register for VAT and can then also reclaim the VAT that they pay on supplies such as building works and professional fees like accountancy - just like every other business. Schools that have carried out a big building project or something in the last few years will probably be able to reclaim quite a lot of money against VAT paid on fees (unless this is excluded in the legislation).
After school clubs are not education and not schools. There will not be VAT on after school clubs run by private companies, unless their turnover exceeds the VAT theshold - some probably already do exceed it and therefore already charge it if they are large companies running clubs in lots of venues. Same with tutoring or dance classes or whatever. Unless the Labour party decide to make all after school clubs subject to VAT, the position will be unchanged.
As it goes, it is actually state schools/LAs who are currently getting a tax break to some extent since although the schools cannot reclaim VAT, the LAs can (which is lucky since their budgets are very squeezed).
It is a pretty ill thought out plan, to be honest and I am not a fan as the amount of money raised for state schools will be negligible compared to what is needed and it may even backfire as people find the extra money unaffordable. But it won't impact childcare according to what has been proposed so far, which is what after school clubs are rather than education - these companies' eligibility for VAT will remain entirely based on turnover, which is already the case.