I believe my studentship is from my department and not a research council (if this is possible). Will I have to pay fees?

You need to check the terms of the studentship, but the usual arrangment with department funding is a fee-waiver plus a bursary for living expenses, in return for teaching duties. Its highly unlikely that if they are funding you, they are going to charge you fees.

It's likely that your fees will be covered by the department. However, if you are from outside the EU you may have to make up the difference between the home and overseas student fees yourself. Check with the department as the rules for departmentally funded studentships are set by the department - they will be happy to tell you where you stand.