It won't mean a significant increase in fees.
The value of charitable status is about £200 per pupil per term (source: The Guardian). That's less than the variation between fees between schools and of course wouldn't apply to those schools that are businesses.
I have never seen a full proposal for removing charitable status. At present, charities have to be wound up in accordance with the law, which means that all assets must be sold at proper market venue, and the proceeds donated to another similarly-aimed charity. That's going to mean closures, and I really wouldn't want to have a child in either sector as state schools have to accommodate those left with no school as a result of closure. Because they're not awash with money to create new places.
If Labour really is looking at this, I hope they publish an exact proposal soon, as this is very much a devil-in-the-detail issue and the consequences of getting it wrong will affect all school age DC.
Imposing a form of tax on school fees is a totally different question, unrelated to charitable status (currently exempt as still the same as the EU regulation which is no taxation on nursery, school or university fees - same one covers all those). Post-Brexit it is however something that can be done.
I really hope it would be done as part of a proper review of successor VAT, not piecemeal (unintended consequences also possible, and I don't trust any party to be competent). But I think this tax change is far more achievable, and likely to raise considerably more tax revenue