As long as you are booked on one ticket through from the UK to NZ, your allowance will be dictated by the operating carrier for the longest segment. It’s called the IATA Resolution 302 Most Significant Carrier rule. Adding excess baggage is very expensive because of the number of legs you have, I would guess up to GBP500 each way.
I am aware shipping can cost less if you are very organised and use ‘by sea’, but it can take 1 month or more, each way, so you would be without your bike in the UK either side of your trip.
Two other options:
1. Upgrade just part of your flight. If you can upgrade at least one leg (segment) of your trip outbound and return to Premium Economy, under Qantas rules you would get 40 kg. You would have to ring whoever issued your ticket to see if you can make any changes and how much it would cost.
2. If the longest leg is with Qantas on both the outbound and the return, you can join Qantas Club for AUD720 which will give you 42 kg checked baggage allowance in economy. It will also give you + 1 guest entrance to the business class lounges on flights booked under the QF or Jetstar flight number (if your flights are operated by Qantas but booked under the EK flight number you would get the extra baggage allowance but not the lounge access).