The people looking to move to colonies from Britain were typically those who were willing to risk the long sea voyage and the hard grind for a chance to build a better life. And English companies set up to colonise NZ focused on bringing large amounts of the working class so that they could do what was needed to make the "estates of gentlemen" productive.
But then combine the gold rushes[1, 2, 3], the Scottish colonisation of Otago/Õtākou[4], and other people settling who had no ties to the English class system (Irish via Australia, Chinese gold miners, Scandinavians up in Dannevirke), the the English class system was never going to really take hold, although the people who formed the new "upper classes" in terms of property holdings and social status did try to maintain it.
And all that said, the Canterbury Association that formed Christchurch did its best to ensure that Christchurch was the most "English city", it was strongly tied to the Anglican church which was gifted rather large plots of land (as was the Anglican private school, Christ's College[5], still a large landlord in the city), and indeed, the Canterbury diocese still holds a rather large property portfolio[6].
Christchurch is probably the most likely place for people to ask which school you went to, and where going to the same school as your interviewer might open some doors, but it's a lot less prevalent these days, IMO.
[0]: https://en.wikipedia.org/wiki/Gustavus_von_Tempsky
[1]: https://en.wikipedia.org/wiki/Coromandel_Gold_Rushes
[2]: https://en.wikipedia.org/wiki/Otago_Gold_Rush
[3]: https://en.wikipedia.org/wiki/West_Coast_Gold_Rush
[4]: https://en.wikipedia.org/wiki/History_of_the_Otago_Region#Sc...
[5]: https://en.wikipedia.org/wiki/Christ%27s_College,_Christchur...
[6]: https://www.stuff.co.nz/business/industries/89760490/churchs...
Some families in Christchurch are very proud of being descended from "First Four Shippers" (colonists who arrived on the four ships that started the settlement of Christchurch), and they all try to send their children to the "right" schools (private ones), but the old boys' club is far weaker here (seems to exist mainly in law firms).
And no-one has class related accents (although a private primary school called Medbury, is very proud to teach a particular accent to its pupils, but they typically have it bullied out of them within a year at high school.