If there is no Will, the estate is distributed in accordance with the Administration Act.
The Public Trust can advise on these issues: website

Who has the legal right to make decisions about the deceased’s body?
- from www.communitylaw.org.nz (edited excerpts):

The deceased may well have made clear what they wanted to happen, either in their will or just by telling family and friends, and in that case their wishes will usually be followed. If the deceased didn’t tell family and friends what they wanted, then the family will decide together what is to happen.

If the deceased didn’t leave a will, or if the will is invalid or didn’t name an executor, then the decision-making power usually lies with the closest family member, as the person with the best legal claim to deal with the deceased’s property. When there’s no will or executor, the deceased’s property is dealt with by a relative appointed by the courts as the “administrator” for the estate. There’s a legal order of priority for who the courts will appoint for this: at the top of the list is the spouse or partner, followed by the children, and then parents, sisters/brothers, and aunts/uncles. However, you don’t have to have already been appointed administrator to be able to exercise decision-making power about the deceased’s body – what counts is that you’re the one who has priority.

Takamore v Clarke [2013] 2 NZLR 733 (SC); High Court Rules 2016 (LI 2016/225), rule 27.35