75% of the population will have access to fibre at gigabit speeds by 2019 [1]. Coverage will increase to 84% by 2024. Not sure about latencies off the top of my head, but there are several new international cables under construction [2] which will help which accessing sites hosted overseas.

[1] https://www.crownfibre.govt.nz/ [2] http://www.hawaikicable.co.nz/

If you find somewhere with UFB access (you can check the maps here: http://ufb.org.nz/maps/ ) then you'll be able to get up to 1000mbit down/500mbit up with a choice from 10+ ISPs. UFB is available in most of the main centers now, though there are still frustrating gaps in coverage, and then you'll be on some variant of ADSL. There is talk of upgrading the UFB technology from connections from the current 2.5Gbit GPON to 10G-PON also, so they'll be able to offer even faster connections.

Latency and international bandwidth can be bigger issues, US is between 150-250ms, Australia 60-120ms, UK ~300ms. It actually has a pretty significant impact on how 'snappy' browsing feels - Facebook, HN etc feel faster in the US even on a slower connection.

Server-grade/Datacenter transit is very expensive here, especially for international bandwidth. Many of the hosting/VPS companies here (like sitehost.co.nz ) have unlimited national bandwidth and then very small international caps.