It doesn't state anywhere that you need to have lived in the UK for at least three months before they would grant you an NI. I spoke with DWP and they said they do prefer if you have an address in the country, doesn't necessary need to be yours, a correnspondants' would do, but it would make things easier.
Otherwise should your employer be able to help you. DWP has a fast path application process for employers and you won't even have to attend an interview.