I assume the OP is trying to make a point about levying VAT on private school fees by saying universities should levy VAT as well.
As several posters have pointed out, VAT is a tax and is paid to the government by the collecting entity. The university would not keep any of the money.
So unless your proposal is that they charge VAT, hand it to the government and then the government pays it back to the universities, then charging VAT on uni fees would not increase revenues for universities. All the VAT goes to government.
If you do propose that the government then pays the VAT back to the universities, it would be much more efficient to raise fees by 20pc rather than go through a process of paying it back and forth.
But also universities are regulated in terms of the fees they charge and there is no state provided free option so consequently they aren't considered a luxury good, like private primary and secondary education is.
Some universities are struggling because they followed government encouragement to increase enrollment of international students, who pay higher fees and subsidise domestic students. Now they have deterred international students through the visa rules and work after graduation rules, the universities which expanded expecting those students are now struggling.
They did it because the government encouraged it through the way higher education is funded. They were responding to the incentives provided to have more international students and now they cannot.