There are a number of significant threats New Zealand (like all the other nations on the planet) faces from space.
1) Intercontinental Ballistic Missiles ICBMs and Submarine launched Ballistic Missiles SBMs
2) Bolides from large Near Earth Objects
ICBMs and SBMs are weapons which can be used without any conventional support. Warheads typically carry thermonuclear or biological weapons. The main threat is the Russian R36-M2 (SS-18 Satan) which is the only Russian missile able to reach New Zealand. One missile could devastate New Zealand's largest cities. However ICBM strategy dictates that all ICBMs are used to attack enemy nations with ICBMs (ideally their ICBM launchers) so that the victor ends up with all the remaining nuclear weapons and the loser has none. Given this strategic necessity wasting a missile on New Zealand (which is no threat) is most unlikely.
Bolides are a significant threat to the whole planet but are not a defence matter. Civilian monitoring is useful but some objects can still surprise.
Geomagnetic storms could severely attenuate all radio communications causing numerous problems. Fortunately civilian space monitoring of the sun makes prediction possible.
Threats to NZ >