The rule people get wrong
The requirement is driven by where you have lived, not by which passport you hold.
That single sentence explains most of the delay in this area. Someone with an Indian passport who spent four years in Dubai and two in Singapore may need certificates from three countries. Someone with a British passport who worked in Germany may need two. Seafarers, contractors and anyone who has followed work across borders routinely need more than they expected.
Immigration New Zealand sets the length of residence that triggers the requirement, and the age from which it applies. Both are on their police certificates page and both are worth reading rather than assuming, because people misremember them in both directions.
Certificates expire
A police certificate has to be recent when it is submitted, and it stays valid for a limited period afterwards.
This creates a genuine trap. Get them too early and they can expire before the application is decided, particularly if there are delays. Get them too late and the application waits. The workable approach is to start early on the countries you know are slow, and to check the validity rule before submitting anything you obtained months ago.
Every country issues them differently
There is no common process. Some countries issue them online in days. Some require fingerprints taken in person. Some will only send the certificate to the requesting authority rather than to you. Some require an application through their embassy, and some have no functioning process at all.
Immigration New Zealand publishes country-by-country instructions naming the issuing authority and the process. Use that list rather than instructions from an agent or a forum, because it is the version Immigration New Zealand will accept and it is updated when countries change their systems.
Where a country genuinely cannot issue one, there is a process for explaining that. It requires evidence rather than an assertion, and it is worth raising early rather than at the end.
Start these first, not last
Police certificates and medical examinations are the two documents that come from third parties on their own timetable, and together they account for most of the delay in otherwise complete applications.
They are also the two people leave until last, because the rest of the application feels more urgent. Reverse that. Begin the certificates on the day you decide to apply, and assemble everything else while they are in progress.
If you have something on your record
Disclose it. Immigration New Zealand frequently has the information already, and a conviction that is disclosed and explained is an entirely different proposition from one that is found.
Character issues are assessed, and not every conviction is fatal. What is close to fatal is being found to have concealed one, because that becomes a credibility finding that attaches to you across every future application, in New Zealand and in other countries that ask whether you have been refused a visa.
If your record has something on it, that is a good reason to get licensed advice before you apply rather than after a letter arrives.