Sarah, I don't know if you have resolved this but I have recently moved over to Brisbane and the transferring of licences is a joke in my opinion. I have a full uk licence including motorbikes, coaches, mini busses HGV class 1, forklifts and dangerous goods. The only licences you can transfer into Australian licences are your car and motorbike licence. Everything else you need to retake a driving test again, at your own cost, I may add.
There is no such thing as class two here. You have LR, MR and HR licences: Light Rigid, Medium Rigid and Heavy Rigid. I think that HR would be the closest to class 2.
Going on from that, you have HC and MC: Heavy Combination and Multi Combination. HC is the equivalent of Class 1. The UK has no equivalent to MC. That is road trains and such like that. I have attached a couple of pieces of information on the categories of licences and Heavy Vehicle Descriptions. There are so many, I wouldn't have the space to write them all down. If you need any other information, let me know and I'll try to help as much as I can.