While this may not apply to you as holder of an Oz
PR, CASA can only start the process of issuing the ASIC to foreign nationals once they have actually (legally) entered Australia.
And for that very reason, DO NOT apply for your ASIC through CASA.
The only check that has to be done upon arrival is the DIAC check and that can be done online by the issuing body.
Go to
Aviation ID Australia for your ASIC. It is possible to have all the documentation completed and the ASIC prepared for prior to your arrival pending the final immigration check.
The best time I heard for somone getting a PPL done full time was about 4 - 5 weeks but that was before the current problems of ASICs and CASA bureaucracy.