Hi,
sorry, don't know much about visas etc. But everyone who wants to work in NZ, needs a permit, ecen as a volunteer. Calling the NZ office in London sounds like the best idea to get information.
Yes, I would contact the DHBs etc directly. First, however, you need to decide, where you want to work. Once you decided on the city, get into contact. Would wait for that though until the beginning of the year. Those jobs will come up spontaneously and they won't be able to promise you anything. Let them know that you are available and what your skills are.
Cheers,
Fyzzio