I think most of the 'exodus' will be in the form of families who choose not to start private (or do fewer years) rather than existing families who leave - although some who did earlier years might leave at a sensible transition point like Y7 or Y12.
So in many places it won't be obvious: except that private school numbers will shrink more than state school numbers over the next 10 years or so. (In a few places - like Edinburgh and Bristol - the numbers may be high enough to be very obvious unfortunately.)
Many who choose never to start private will keep one parent working part-time instead of full-time. Lots of families have the Mum working 3 days per week from when the kids are born and taking kids to activities (I'm sure you know some. It's nice: less stress for the whole family). Private school for 1 child is about the difference between a parent with a professional job (like doctor, dentist, accountant, solicitor) working 3 days/week vs 5 days/week. So going full time to fund private school is currently an option for those parents, especially since private schools facilitate full-time work with longer days and extracurriculars available at school. So if they don't make that choice (ie to increase hours to fund fees) then they won't actually have any 'extra' money in their pocket to spend: they just won't shift back to full time. They might not even realise it's VAT which has taken away the option. The big downside for the government is that they won't get that extra £13k per year income tax (as well as the government now having to fund the DC's education). And there usually aren't enough people with those skills as it is, so the job won't go to someone else: the work just won't be done, with impacts on the economy/growth as well as income tax.
Others - maybe those who prefer working full-time anyway (I'm sure there are some!) or are saving mutiple kids fees - will choose to put extra money into their pension. That actually has income tax consequences both immediately and also because a bigger pension will allow them to retire earlier: reducing income tax at a later date too.
Or buy a bigger house. Or one in catchment for a better school. Especially over the next 10 years or so, as people with kids just being born now make different decisions about their medium/long term future. So the government may get say an extra £20k stamp duty per family (total) in some areas with good state schools (family home in the £1mil band, prices increasing £200k due to increased pressure on catchment). But I suspect the £20k stamp duty going to the government now may be offset by lower income tax in 20 years when they downsize and use the house price growth to retire earlier.
There will probably be some extra spending on goods, maybe cars and electronics, which does attract VAT. Personally I don't think it will be a high proportion of the 'money not spent on fees', but that's a gut feel based on my own priorities: it will obviously depend on the family.
Holidays are likely to take more of the 'saved fees' than consumer goods in that income bracket I think (although that might just be my own preferences!) but still not as much as reducing work and investing in pension. Holidays don't attract VAT but would at least be neutral for income tax (ie are still money spent, which they would have paid income tax on - like fees).
None of those different outcomes seem worth the downsides and the risk. If 10% of the children who were in private schools or would have gone private go to state schools instead, the government will have less money than if they hadn't added VAT to education. So they will have to cut some other state spending they could otherwise have paid for. That's bad for everyone: whether it happens next year or whether it gets progressively worse over the next 10 years as the change filters through.