The India side
In India, interest on an NRE account is exempt under Section 10(4) while you hold non-resident status, so India charges nothing and deducts no TDS. NRO interest is taxable, with TDS under Section 195 at about 30% by default, which the India-New Zealand treaty caps at 10% if you file a tax residency certificate and Form 10F with the bank before the interest is paid. So on the Indian side, NRE is untaxed and NRO is taxed at up to the treaty rate.
New Zealand, the trap, and the window
New Zealand taxes a resident on worldwide income, so both NRE and NRO interest are taxable there. Plain bank deposits sit outside the deemed-return fund rules that catch foreign shares, so this is ordinary foreign interest income. The trap is on the NRE side: it is exempt in India, but that is an Indian rule, and for a NZ resident it is fully taxable, and because India levied nothing there is no India tax to credit. So the tax-free NRE account is tax-free only in India. On NRO interest, NZ taxes it too but gives a credit for the India tax, up to the treaty rate of 10%, relieving most of the double charge.
The timing exception applies here as well, and it is generous. For roughly your first four years as a new NZ tax resident, the transitional-resident exemption makes both NRE and NRO interest exempt in New Zealand. So a recent migrant genuinely keeps the tax-free NRE benefit during that window, because neither country is taxing the NRE interest, India by exemption and NZ by the transitional relief. Once the four years end, the NRE interest becomes fully taxable in NZ with no credit, so it is worth planning the accounts around that cliff.