As JandM says, everyone has a different experience, there really is no exact answer.
For us we were pulled out of the pool but had to wait a month before we received the "congrats, you've been pulled out the pool, now we need..." email.
Searching "timelines" on this site will give you an idea of how it was for others, it might be quicker or slower for you but at least you get some ideas.
Back to the job hunting. I wouldn't rule out any company yet (especially one having such growth). I think the first challenge is to get a job for local experience, then you can hunt around for the job that fits your aspirations better.
If they have a phone number to be honest I'd give them a call. People are so friendly here, call up, explain you're looking for a job as X and just you'd like more information about that position in their company. Even if you don't want to apply (timing not right or for other reasons) or even if they don't have a position today, get to know them and what the role really looks like here. Perhaps you could ask to meet someone currently doing that role for a coffee.
Seriously NZ is small, whichever city you end up in people within the same fields know each other, so start getting to know them
Hope it goes well