The legal position of the former British dominions, Canada, Australia, New Zealand, South Africa, and Eire in their relationship, with the United Kingdom:
The 1926 Imperial Conference devised the Balfour formula of dominion status, stating that:
    The United Kingdom and the Dominions are autonomous Communities within the British Empire, equal in status, in no way subordinate one to another in any aspect of their domestic or external affairs, though united by a common allegiance to the Crown, and freely associated as members of the British Commonwealth— Balfour Declaration of 1926 
The Balfour Report further resolved that each respective Governor-General occupied “the same position in relation to the administration of public affairs in the Dominion” as was held by the Monarch in the United Kingdom. The only advisers to the Governor-General and the Monarch in New Zealand were his New Zealand Ministers.
and
https://en.wikipedia.org/wiki/Statute_of_Westminster_1931
I quote:
	
	
		
		
			The Statute of Westminster, 1931 is an Act of the Parliament of the United Kingdom and separate versions of it are now domestic law within Australia and Canada; it has been repealed in New Zealand and implicitly by subsequent laws in former Commonwealth realms. Passed on 11 December 1931, the act, either immediately or upon ratification, effectively both established the legislative independence of the self-governing Dominions of the British Empire from the United Kingdom and bound them all to seek each other's approval for changes to monarchical titles and the common line of succession. It thus became a statutory embodiment of the principles of equality and common allegiance to the Crown set out in the Balfour Declaration of 1926. It thus had the effect of making the Dominions sovereign nations.
		
		
	 
It is noteworthy that on the abdication of Edward V111 the British Government was obliged to seek the approval of the then dominion governments to enable George V1 to become King. On matters of imperial defence all the dominions, and the UK had an equal voice when debating mutual assistance during times of war. It is no secret that the UK, NZ, Aus., Canada, and the USA continue to work closely on matters of common defence. 
Further, following the decision of the newly elected (1945 general election) Labour government to dismantle the empire, consultation was set in motion with dominion governments on the issue of British nationality as it pertained to British subjects living outside the United Kingdom, and her dependent territories.  United Kingdom legislation that changed British nationality law, was reciprocated in former British dominions.
Transparently the majority of the European population in New Zealand are descended from settlers who had emigrated from England, Wales, Ireland, and Scotland making it easy to understand why many Kiwis treasure their historical relationship with the United Kingdom, and Ireland.
Here's an article detailing in some depth the matter of New Zealand's nationality legislation:
https://en.wikipedia.org/wiki/New_Zealand_nationality_law