Would they just cancel the whole thing? Would they arrange for an overnight transfer for me?
There are a range of factors that play into this. Firstly if the change is made then they'll attempt to rebook on the next best option which may include something the day before (keep an eye out on the app as schedule changes appear in the app first before an email is sent). If there isn't a next best option then they'll just issue a credit cancelling the booking but you don't need to take this option.
I had a few CHC-OOL and OOL-CHC flights this year where there wasn't any next best option (they weren't operating anything to/from OOL) but I called, was able to have the booking reinstated and placed on a CHC-AKL-BNE option instead - I did have to suggest this though. Mine didn't need accommodation because it was a same day transfer.
Now if they offer accommodation depends on a couple of things. For any changes made more than 14 days prior to departure they won't offer anything.
For changes that are made within 14 days of departure they'll offer a single nights accommodation but they specifically note:
Please note, accommodation will only be provided where the situation is defined controllable, for uncontrollable situations such as Government imposed travel restrictions as a result of COVID-19, accommodation will not be provided.
Now the cancellation of WLG-AKL is very much an Air NZ controlled situation, they're choosing to operate less flights because of depressed demand not because the government is tell them they can't. Despite this early in the pandemic Air NZ was claiming it was an uncontrollable situation for any flight reductions, more recently someone took them to court about it and won so they seem to have changed their tune.
The most recently example is when the NZ went into lockdown and people were still returning from the Cook Islands. They still operated one way flights from RAR-AKL but due to complete lack of domestic flights many people had to wait in Auckland until the next day, in this situation Air NZ did pay for a nights accommodation in Auckland for those impacted.
Their official policy outlining this is listed here, the page intended for travel agents but the same policy applies to the public :
Schedule Change Expenses
I guess it'll mean if a night is required and they change the booking within 14 days of departure then they should pay however you may have to be prepared to quote their own policy and get them to honour it.