how do! We're planning to book our flights using BAMiles (different to AirMiles), also collected via Tesco clubcard points.
We have seen in the past that it's difficult to get London-Sydney tickets (we're 2 adults, 1 child), but if you tick a box to use other carriers you can fly via other OneWorld airlines. Flying via Frankfurt via Qantas (including a short London-Frankfurt hop on BA CityFlyer) didn't alter the number of BAMiles needed, though the taxes did go up.
We're also collected a bundle of StarAlliance miles using a BMI credit card, which will take care of the Sydney to Christchurch journey (other than taxes, of course).