You live and study in the UK and according to your passport you are a British citizen (presumably you were born of British parents?)
You don't need a work permit to work in your own country.
Your employer may need to have evidence of your right to work as this is a duty placed on them under government rules and this would usually be in the form of seeing your passport and taking a photocopy in case they ever get asked by the authorities.
If you don't have a passport, then you can provide a copy of your Birth certificate PLUS a letter from the Department from Work & Pensions that has your National Insurance number on it.
At your age there are rules over the number of hours you can work in a day, the rest periods you must be given and so on and these are designed for your protection.