Is it my responsibility to contact the police to see where it is?
There is no legal responsibility but I get the curiosity.
Or can it take several months for them to issue the NIP to the named driver?
Usually weeks, not months. Given they have 6 months to prosecute the underlying offence it would be monumentally stupid for them to wait so long (but it does happen).
Arguably there's nothing to be lost in trying to contact them (Although there's an extremely slim chance they do have time to get a s172 to you and prosecute depending on the exact dates). But as you don't have any paperwork that may be a challenge - although Googling the local safety camera unit may produce something.
There's little point commenting further without knowing the hand you've been dealt.