The title of this thread is a little misleading.
Let me clarify conditions in Denmark,
Prince Christian and Princess Isabella the two eldest of Crown Prince and Crown Princess' children are both at secondary schools and it is correct that the school Prince Christian goes to is not a fee-paying school, but is government funded, except for such activities as school trips abroad and the like, where parents have to contribute to the costs involved.
Princess Isabella is at a private secondary school where parents pay 1700 D. kr. monthly is fees. This at today's rate of exchange is £192, 92.
It is correct that their younger brother and sister are at a state school, just as the two elder children were for primary school.
Their father and grandmother were educated at home for part or most of their primary education, then sent to private schools in Denmark and later in France for Crown Prince Frederik and his brother, while their mother , the present queen, went to a private school in Copenhagen, a boarding school in England and later to university.
So the transition towards educating royal children here in ordinary schools started sligthly before it did in Britain.
A factor that should also be mentioned is that Denmark has relatively few private schools and most of them receive very much larger amounts in state funding than any private school I have ever heard of in the UK, so school fees are much lower here for that reason. The few private schools that have no funding, do not qualify for it because they infringe ministry of education guidelines in some minor way. Major infractions would be grounds for closing a school here.
For instance, a private school cannot use coporal punishment as this is illegal in Denmark, but a private school could insist on segregating boys and girls, or require girls to wear hijab - but will not qualify for state funding.
Likewise a Christian private school that insists on teaching that the earth is flat, or that Darwin's theory of evolution is blasphemous might be allowed to continue so doing, but not for the tax-payers money.
These days, unfortunately, the royal children here too have to be accompanied by security details. This is inconvenient for them and the school, but is no greater incovenience in a state school than a private one.
Prince Joachim's children have likewise alternated between state schools and private schools.