"When would you be able to start" is a fairly common interview question, especially towards the final interview - the employer wants to know if you have any notice period, travel plans, or contractual obligations that prevent you from starting immediately.
In your case, your desired start date is much later. I think the professional thing to do would be to bring it up yourself at the first opportunity - at the end of the interview when they ask "do you have any questions for us?" They probably already know that you are a student, so it will not be a huge shock.
To show that you are willing to be accommodating if they want to play ball, you could communicate your willingness to attend training for the position slightly earlier than your official start date (for instance, if your school is local, and you can come in on weekends or days during the exam period when you don't need to study).