Repeating some of the above, but just to clarify, under EU law a member state must set the same level of University tuition fees for the students of all members. Not only that but they must make the same provision for paying for these fees - so EU students are entitled to tuition fee loans in England and to free tuition in Scotland.
It works the other way around. UK students may study in other member states' Universities on the same terms and at the same cost as their own students pay.