All the schools will need to register for VAT.
When they issue an invoice, they will add VAT to it (so the total paid by parents will be fees + 20%). This will mainly be invoices issued to parents for fees, but if they charge eg other schools to use their facilities those invoices will also have VAT added.
VAT is due every 3 months. When the school do their VAT accounting, they'll count up how much VAT they've received from parents/customers, then they'll subtract how much VAT they've paid on things they've bought (school supplies, vehicles, services they buy in like cleaning etc), and they'll send hmrc the difference.
The subtracting of VAT they've paid is why the VAT might end up costing parents a bit less than 20%. Because private schools aren't currently able to claim back the VAT they have pay on things they buy (state schools can). So they'll pay hmrc a little bit less VAT than they receive, which may allow them to not increase fees as much as they would have otherwise (increases reflecting ever-rising costs).
There's a bit of complexity when they first become VAT registered, where they can claim back the VAT not only for that 3 month period but also for things they recently paid VAT on: within the last 6 months for services and within the last 4 years for goods.
So they might be able to offset a reasonable amount of VAT initially if they've had building projects etc. But over time, it won't reduce the percentage very much since a school's main cost is salaries, which VAT isn't paid on.
I suspect the government won't wait until the VAT receipts start rolling in to spend the money.
The bigger challenge for them will be finding 6500 extra teachers to recruit, given that there are currently 2000 unfilled teaching vacancies. The breakfast clubs will hopefully be easier to implement.