I moved to Australia last year and am going through this process, the changes that were made (late last year I believe) mean that you can convert a UK to CASA PPL without having to do any exams; you get issued the license and then need to do a Flight Review with an instructor.
Process is as follows:
1. Apply for an Aviation Reference Number - CASA Form 1162 . This part is free, but needs to be done before any other steps.
2. Find a Medical Examiner (DAME) and take a Class 2 Medical
3. Provide authorisation to the UK CAA for License Verification, form SRG 1160. This is so that CASA can verify you're license is genuine, however you will need to send a certified copy of your license for this, see the form for who can do this.
4. Obtain certified copies of the following documents: Certified Passport Photos, Copy of License, 3 last pages of logbook, Australian 100 point ID check docs.
5. Complete Form 498 (Australian Security Identity Card), Form 61-9PIC (Photos), Form 61-4A (Flight Crew License Application on basis of Overseas Civil Qualifications)
6. Once you receive your license, you need a Flight Review (Biennial Flight Review).
Must be done in that order.
I called today to follow up, and although the CASA licensing department have been helpful with all of my phone calls, they did advise it will take up to 56 days to process the license application, after which it goes to a separate office where the process the ASIC which takes up to a further 4 weeks.
Next I just need to find some flying buddies and a group share at Camden!