Hi Annie,
You do have to register with AHPRA, your APC only allows you to practice in New Zealand just as your HPC registration only allows you to practice in the UK. AHPRA is Australia's equivalent of the HPC, you need to register with them to be able to legally work as a physio in Australia, you wont be able to find work here without it. All the states have now combined so once you're registered with AHPRA you don't need to register with individual states (at least not Queensland, Victoria or NSW).
You don't need to be a member of the APA. They're a little bit like the CSP and you do get discounts on courses if you're a member but unlike the CSP you don't get any form of insurance from them. If you're planning to work in private practice you will need professional indemnity insurance and public liabilty insurance to cover yourself. You can buy this online and it comes through in minutes so you don't need to worry about that until you have a job offer (some private practices will even pay for it for you if they like you enough).
In short AHPRA is Australia's version of the HPC, you need to register with them. APA is like the CSP but without the insurance aspect, you don't need to register with them.
Hope this helps.
Claire