It has been talked of Nurse Practitioners having a role on the specialist low-acuity response cars (Sierra/UCC) but this hasn't happened yet in the approximately five years it has been running and is unlikely to happen in the near future.
There are only a tiny number of NPs around (the Nursing Council puts it at 142 vs 47,488 RNs in 2015 - or 0.002% of the Nursing workforce) and I bet not very many work in an appropriate acute setting to transition properly e.g. ED or an A&M clinic.
Physician Assistants (called Physician Associates) have been trialled twice - the Ministry of Health is not interested in pursuing them further and honestly I do not think it should be really. There was some grind from the Junior Medical Officers about them working in acute care and that is fair enough. They would be great in the smaller areas where attracting GPs is proving difficult, and will forever be difficult I suspect. That however, is another topic.
To be fair, no I do not think the value is worth the hassle; the vast majority of what an NP would do is already undertaken by ambulance personnel in terms of making a diagnosis and plan of action for what that patient needs; if I get on the blower and give their GP a swift one they'll generally be seen the same day especially if they have a good history, otherwise if it's just an acute problem the local A&M will always see them on a walk-in basis. This takes care of the "other half" of the problem in that ambulance personnel cannot prescribe, and nor do I think it should be.