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/