Wow, the IRD is really hard to get in touch with if you don't have a landline phone (although, haha, even if you do sometimes). I tried calling them this morning, and get the usual message of "higher than normal wait times, but we can schedule a call back". Only the call back option doesn't accept mobile numbers, and there's no option just to wait on the line at your own expense.
The only way I could get through to someone, was to ring complaints, and they'd transfer me.
If anyone else gets stuck with the same problem, the complaints line seems to be a quick way through.