ICAO has the following courses:
Air Traffic Control Assistant/Basic Induction (ICAO 051)
Aerodrome Control (ICAO 052)
Approach Control - Non Radar (Procedural ICAO 053)
Area or Approach Control - Radar (ICAO 054)
Area Control - Non Radar (Procedural ICAO 055)
The content of these courses differs from the European Common Core Content (CCC) ATM Courses. CCC courses are basically built upon the ICAO courses, but are further developed.
Individual countries' Aviation Authorities approve courses (own national training agency courses or CCC courses for example) and or acceptance of other qualifications (other country training agencies).
Therefore an ICAO course, although a good worldwide base, does not guarantee employment globally, as a nation's Aviation Authority may not accept it, without extra training.
Also just because NZ people may give the tuition, they will have to provide ICAO content, not NZ course content, therefore your qualification may not be accepted by NZ Aviation Authority, without extra training!!
Hope that clears it up a bit for you.
Just jump onto google and start doing some digging and reading!!