When a marriage breaks down and there are children involved the court’s primary consideration is their welfare and needs. If one parent is taking the majority of responsibility for the childcare then the other parent must make payments towards the upkeep of the children.
Child financial provision is usually decided by the court at the same time as the other finances are decided. This is generally the case if the parent not taking the majority of responsibility for the childcare will probably be paying over the minimum amount of support required because they are more wealthy than the parent that is responsible for the majority of the childcare. This is when who pays how much of the school fees will be decided.
The court will look at factors such as whether the child already goes to a private school or it was expected that they’d go to a private school. Another factor the court could take into account is that private school is more common in England than in mainland Europe.
Alternatively, you can make an application via the Children Act.