What is the rough salary one should expect from a research university in New Zealand, say in mathematics or sciences areas, at the beginning assistant professor (called Lecturer in NZ) level professor?
You can find the current collective agreements for all universities on the Tertiary Education Union's website. Entry-level salaries for a lecturer, and the low point of the senior lecturer range, at the start of 2021 were as follows (all NZD):
|University||L1 salary||SL1 salary|
The average is about NZ$78,000. These generally have annual percentage adjustments built in, so the base rate increases each year. There is generally also an automatic progression within a grade for current staff across several steps spread between the lecturer base and senior lecturer base, and it is generally possible to be appointed mid-grade as well according to experience (e.g. working time since PhD). All of this varies from school to school, and there are also other payments that may apply, so you should check the agreements for particular details you're interested in. If you're not a union member these rates will generally apply to you when you start as well (by law, historically, but the situation may change).
A couple of the agreements linked above have expired, and may have been replaced but not yet made available on the website, and some contain increments that hadn't come into effect yet. The general area of the numbers is right, though. There is a fair amount of variation between institutions, particularly the largest and smallest universities. There's also quite a bit of variation in how the scale works above that lowest-level entry point and how many steps there are (between 6 and 9 before moving to SL).
Wikipedia suggests that "assistant professor" is equivalent to "Lecturer" in the Australian and New Zealand Academic ranking systems.
My guess is that you can google any New Zealand university and they will list salaries for a Lecturer. For example, I searched for "university of auckland academic salaries" and obtained the following document
So if you were employed as a lecturer in 2014, you'd probably start at $76,535 (New Zealand Dollars; exchange rate as of 19th Dec 2014 is 1 NZ dollar buys .78 US dollars; so starting salary would be around $60,000 USD) and each year you'd typically go up an increment (i.e., about $2,500 NZD; $1,950 USD) until you hit the top of "lecturer". At that point, further pay rises would be contingent on applying for a promotion to senior lecturer.
The relevant section from the University of Auckland agreement is shown below.