Wellington Regional Emergency Management Office has advised the following re why it can take some time for Tsunami warnings to be issued. :
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After an earthquake, GNS Science and the National Emergency Management Agency (NZ Civil Defence) conduct an assessment on the earthquake’s location, magnitude and depth. This information is used to determine whether or not the earthquake has the potential to generate a tsunami that could affect New Zealand. This assessment takes time and sometimes an earthquake can be complex, meaning it takes longer to determine whether there is a tsunami threat.
For a local source tsunami (which is one that has been generated by an earthquake in or near New Zealand such as in the Cook Strait or Hikurangi Subduction Zone), which could arrive within minutes there won’t be time for an official warning. The earthquake may also have damaged power and telecommunications which would affect the ability to issue warnings. This is why it’s important to know the NATURAL WARNING signs of tsunami and act quickly.
In this case, the earthquake is your warning – If you are in a tsunami zone and FEEL an earthquake that is either longer than one minute or so strong standing is difficult, you must evacuate immediately to higher ground or as far inland as possible once the shaking stops. DO NOT wait for an official tsunami warning.
For earthquakes that occur further away from NZ (such as in the Pacific Islands or off the coast of South America), tsunami arrival times can range from 1-12 hours. In this case, power and telecommunications would be running as normal and we have more time for assessment and issuing of official tsunami warnings.
Remember, if an earthquake is Long or Strong: Get Gone! The earthquake is your only warning. If you live in a tsunami zone, make sure you have a Grab Bag with essential items you can take with you if you need to evacuate with little or no warning. 
For more information on tsunami, or to find out if you live in a tsunami zone, head to wremo.nz/tsunami