UK [...] does not have a proper written constitution.
I'm not sure NZ can claim a "proper written constitution" either, based on this url:
https://www.lawsociety.org.nz/lawtalk/lawtalk-archives/issue-890/towards-a-written-constitution-for-new-zealand
Excerpt: "the New Zealand constitution was located in 45 acts of Parliament (including six very old English acts), 12 international treaties, nine areas of common law, eight constitutional conventions, three and a half executive orders, one prerogative instrument, one legislative instrument and half a judicial instrument."