OK, I found what RSM UK has to say about it:
"There will be no catch-all exemption from charging VAT for private school fees for children with special educational needs and disabilities. Exemption is limited to ‘non-maintained special schools’ - independent charitable foundations whose pupils are funded through an Education Health and Care Plan (EHCP).
In addition, local authorities and other state bodies charged VAT on school fees they fund for pupils placed at other types of private school in accordance with an EHCP, or similar arrangements in Scotland, Wales and Northern Ireland, will be entitled to recover the VAT from HMRC.
However, despite estimates that only a third of pupils with special educational needs have an EHCP or equivalent plan, there will be no relief from VAT on school fees for children who have such needs that do not qualify for that type of state funding. Parents in this situation will be faced with the choice of paying the VAT themselves or disrupting their child’s education by moving them into the state sector."
The issue is that some children don't have ECHPs. The reason is that there are huge delays in assessments. On the other hand, it's also known that some private schools label some of their pupils as having special needs because it makes it look as though they're doing a better job than might, in fact, be the case. Without an official assessment, it can't be confirmed whether a child really does have special needs - in any case, it's a debatable point what the definition of special needs is. Originally, it mean that a child's individual needs couldn't be met by mainstream education, but it's now being used for a much wider range of pupils - arguably, "every" child has a special need. In some cases, it's the parents who have decided that their children have a special need, so justify sending the child to a private school - those are the ones who won't be exempt.