'Wild Camping' (as per campervan) is essentially prohibited. Camp sites exist but there's a cost at commercially operated sites and and often (but not always) at 'DOC' Department of Conservation sites. See:
https://www.doc.govt.nz/parks-and-recreation/places-to-stay/stay-at-a-campsite/
Advice re: inter-island car ferry cost is well-placed - you will do well to consider Auckland-Wellington (plus wider N. Island touring) as a single leg; ferry arrives into Picton and car rental available there for S. Island touring (likely, you will exit via Christchurch, but there are flights out of Invercargill and Dunedin if you wish to avoid back-tracking up to Christchurch).
As has been mentioned, ANY camping gear that you take needs to be spotless (esp. tent pegs - consider scrubbing and then having them in your checked baggage within a clear plastic bag for inspection); scrub all hiking boots or similar; have photos of the underside of your CLEAN tent to show no debris is present (this may not prevent an full un-packing of a tent but can mitigate the potential).
Do NOT take any fruit into the country: Qantas (and other carriers recently provided passengers with apples, in-flight .....passengers carrying these off the aircraft have been fined on the spot by NZ Ministry of Primary Industries officers at the airport customs clearance. You can declare and dispose upon arrival with an 'Amnesty Box', with no penalty, ahead of Customs. Best to just make sure you leave all un-eaten fruit on-board the aircraft.
Travel to NZ
I have strong recommendations regarding how you might consider your air travel (!) - this may depend upon your age and resilience but in general, please consider a 1-2 day stopover in your route to-
and from NZ. Cost is always an issue but please consider very carefully, the flight logistics of taking the cheapest advertised fare, especially if this involves extremely long flight sectors into NZ (for example, Doha-Auckland, Dubai-Auckland which are each 16-plus hours and jam-packed in Economy); you will arrive exhausted and in very poor condition to drive.
Also, it will take days to fully recover if you are not accustomed to long-haul travel (by which I mean, 12-plus hour flights)..... Remember, the very shortest total elapsed time to travel from a UK airport (plus your travel time from home) will be
26 hours. Even with good stamina, this experience crushes most mortals.
In general, your best routing would be London or Manchester) to Singapore on Singapore Airlines. Stay 1-2 nights in Singapore. Take a morning flight from Singapore-Auckland (Singapore Airlines or Air NZ) and overnight in Auckland (There is a Novotel at the airport, plus other options nearby, but the airport is 10 miles from the city). Try to avoid the overnight flight from Singapore-Auckland as this is not particularly pleasant in Economy. Then start the next morning with your car rental and drive. On the way home, stay in Singapore and relax, then take the 14-hour flight back to UK.
Remember when flying long-haul, SLEEP IS YOUR FRIEND - ignore the in-flight movies and prioritise sleeping. Eat before your flight and ignore the in-flight meals and being woken for them. (For an overnight flight, you can specifically ask the cabin crew to NOT wake you for meals...generally, there are always snacks available in the gally for when you do wake up and are peckish).
I have 20+ years of living there with frequent international long-haul travel, and happy to answer any questions here or via PM.
Good luck with your planning!
