I worked for ~2 months without an NI number. HR kept asking me for it, and I kept putting it off and then finally called and got one (easiest thing I did related to this move, BTW...had I known that, I wouldn't have put it off
). They withheld taxes at the emergency rate before I had the number and then reduced withholdings after that...I do need to claim back the excess paid.
If she can, I would just leave the field blank on the application. If that's not possible, I would but TBD or XXXXX or some such obvious non-entry in the field. Weirdly, for my new job, they wouldn't actually interview me until I could produce my NI number, but hopefully that won't be an issue for you. (For my initial job, no one even asked for the number until I had been working for a few weeks)