HRMC has no record of my employment- what should I do?

I overpaid significantly the tax for my short UK internship in 2025 as the company put me on M1 tax code. I called HRMC today because I thought it could speed the tax refund calculation process but turns out they don’t have me on their record. I’m currently abroad so I’ll have to send in a letter with my P60/P45. What else should I do to ensure I get my refund as soon as possible. How long does it usually take for such cases?