If you buy the USA Nexus 5 model (D820) it will not work on NZ 4G, on any current or future network. It will however work with DC-HSPA+ on all 3G networks.
The D821 is required as this has Band 3 LTE (1800mhz) that is used here. Additionally band 7 (2600mhz) is also supported, that is likely to be used for additional capacity by NZ providers.
Via Anandtech, frequencies added:
(D820) North America: GSM: 850/900/1800/1900 MHz
CDMA: Band Class: 0/1/10
WCDMA: Bands: 1/2/4/5/6/8/19 (2100/1900/1700/850/800/900mhz)
LTE: Bands: 1/2/4/5/17/19/25/26/41 (2100/1900/AWS/850/700bc/800u/1900+/850+/tdd2600 mhz)
(D821) Rest of World: GSM: 850/900/1800/1900 MHz
WCDMA: Bands: 1/2/4/5/6/8 (2100/1900/1700/850/800/900mhz)
LTE: Bands: 1/3/5/7/8/20 (2100/1800/850/2600/900/800dd mhz)
And for future reference, this is largely the case for every other USA device unless it has these bands. The iPad Air is one example of this with a single model that works worldwide.