SQ LHR-SIN-CHC is the fastest. It is also the easiest to get redemption on using UA miles (at least in Y). Getting award tickets (even on Y) on NZ LHR-LAX is almost impossible (I tried it for Easter and couldn't get anything for weeks either side of preferred dates). Search for Starnet blocking in the UA forum for more details.

Also, you get to transit via SIN, rather than LAX or SFO

LHR-YVR-AKL-CHC I would also rate as a good option. YVR is not a bad airport. However, YVR-AKL at 14hours+ is one of the longest non-stop flights anywhere, so bear that in mind (and if you do it without a stopover in YVR, you will leave at ~4am in the morning UK time, which might mean you end up being awake for most of those 14 hours!

One tip, if you wanted a stopover in AKL and to take this route: it is possible to get a <24hour stopover (which is not considered a stopover for the purposes of the "1 stopover max" rule) in YVR if you fly LHR-YYZ-YVR and arrive at YVR less than 24 hours before the NZ flight leaves. Since the NZ flight leaves in the evening, you can easily head downtown for a night in a decent hotel, spend the next day wandering round Stanley Park, then sleep on the YVR-AKL sector better.

But for scheduling, I would try LHR-SIN-CHC, or possibly LHR-HKG-AKL-CHC before going via N. America...