I thought I would post this as it's bound to happen to someone else.
My son is 16 and is now in full-time employment via a modern apprenticeship. He will be earning more than enough to be required to pay National Insurance contributions, but has no NI number.
I called the local Job Centre to schedule an appointment for him to get a NI number (as an adult would do), but they told me that because my husband had claimed Child Benefit for him that he most likely already had an NI number. The Job Centre gave me the number to National Insurance Young People's Registration Line (0191 22 55 670) to see if he does indeed have a number.
I called them and after taking information regarding our address, his birth date, parent names and details of the parent claiming Child Benefit, they lady said that Josh does have an NI number. They will be sending me a form to fill out and return, then we will be issued with his National Insurance card.
No interview, no proof of residency, etc.