Hi Will,
Like your son, I'm a NZ citizen born to British parents so my parents have puzzled over what you're going through before. I don't know if things have changed in the last 20-something years, but here's how it worked for us:
I obviously have a NZ birth certificate, and didn't need a passport until I was 7, at which point my parents applied for one for me- all they had to do to prove my citizenship was to put their passport numbers on my passport application form. This proved they were British citizens, so I was automatically granted the passport with my citizenship listed as British. When I was 17 we ended up moving to the UK for several years, and while there and over 18, I applied for a NZ passport as well, which was easily done as I just had to provide my birth certificate number. Neither passport took long to process (ie, there was no delay because of the dual citizenship). I had two passports for many years, but despite being back in NZ I've let my NZ passport lapse since it's too expensive to have two! For what it's worth, it's easier and considerably cheaper to get a Returning Resident Visa (I think this name may have changed now) from NZ put in your British passport, than it is the other way around (45pounds vs 400). I have found having a UK passport really helpful (which is why I still have that one- they last longer too!).
Again, unless things have changed since I went through this, there's no deadline that you have to get your son a British Passport, unless you intend to go back to the UK for longer than a visit.
Hope that helps!
Ferris