As someone who already was controlling in Australia, and had a Doctor give me appointments quite quickly, it still took CASA 5 months to issue my medical.
I understand now why they are seen universally as muppets by everyone in the industry.
Paperwork lost. Check.
Wouldnt take copies of originals. Check.
Had to get medical redone because of it. Check.
My advice to all undergoing the recruitment process, once you have done your interview and told to do your second batch of testing you more or less have a job, so just go and start your medical and ARN process then, you will have it in time (hopefully) to not delay any course dates.