Caveat - I don't know the answer but the current situation in the UK is this:
According to the HMRC site - especially see "What happens when you apply for a National Insurance number", you should auto receive a National insurance Number when you are 16 (I doubt that happened back then - I can't see how they would know such a child existed in the days before Child Benefit). If not, you have to apply, and you have to provide your birth or adoption certificate or other evidence of identity. That's what happens now. For the life of me, I cannot remember how I got mine. It was certainly pre-1st-Child-Benefit.
Logically, there is absolutely no way that any records for NINo registration of someone born less than 100y ago would be open to any sort of enquiry.
There is, incidentally, the details for the 1939 National Identity Card that are a/v under Freedom of Information requests - for which you do not always need to provide an address. Beyond confirmation of birth-date, this might not be much use for your specific task. Incidentally, the authorities have to check that anyone they provide on your response is dead, so you may get a "bonus" death date as part of the response, as well as the 1939 data.