I don't see that happening.
7% of children go to private schools, it's an exclusive service you opt into when you have the money and feel your children would benefit from this exclusive upgrade (state provision is available to all) I appreciate that SEN children are often underserved by state schools, but for longer term system change those parents need to stay & fight their ground with LAs rather than opt into private provision.
Most children across every social class/income bracket use nurseries for their young children. Nurseries have been shown to prepare children for school and decrease the gap between richer & poorer children.
Universities prepare and train the workforce, yes definitely class/income plays a huge role in higher education and its students, but again there is the promise of levelling up and the loan system is supposed to equal up poorer students opportunity to study for a degree.
I ain't saying it's perfect but I can't see the connection between taxing VAT on private schools = taxing nursery and universities which are not luxury services in our society.
Just as you don't get charged VAT on NHS services, like having a surgery etc. but you would if you used a private provider for cosmetic surgery because that would be viewed as a non-essential.
I get you can be angry about change, people often are, but I don't think it's justified on this issue which has been anomaly and should have attracted VAT a long time ago.