As mentioned above calling the call centre won't really be of much help, You need to simply add your Elite Partner as a giftee on your Airpoints account and lodge the upgrade for them. The call centre could lodge the upgrade for you (you'll have to pay a service fee for this since it can be done online) but won't be able to do anything until you have added them as a giftee.
I would have only waited 5+ mins maybe once or twice in the past year, and one time that was during a busy time of weather disrupts. That message plays a lot even when the premium team don't seem to have a wait.