PRIMARK interview: could I just show them a picture of my national insurance number?

Have they requested a physical copy or document with it on? Usually companies just need you to provide it, verbally or in writing. The companies I've worked for have only ever asked for passport or driving licence for proof of identity, and then I've given them my national insurance number on the new starter form. There's more information on the HMRC site, including this which states you don't need a plastic card anymore. Apply there if you think you'll need something official looking, but my guess is that a verbal confirmation is all you'll need. Good luck!

To be honest when I had my Primark induction they were really strict and wanted everyone to have the correct documents and some people were even sent home for not havign them. I remember a lady in my induction didn't have proof of NI but knew it but told her she needs to bring in paper proof next time she was in.

You might be able to get a new NI letter on the gov website