We’ve not had any posts about this from Australia that I can think of, so can’t give you info specifically about your situation… Would your other doctors be willing to email Prof Elliott to give updated info about your condition as you’re getting worse? That might help push you up the list? As others have said on here “it’s the squeaky wheel which gets the oil” , so you could ring his office/ secretary to politely ask what the system is & waiting times? Also in the UK, most doctors who work for the NHS (free treatment) also do private work which you can pay for to see them quicker, is that a possibility in Australia? It just helps jump the queue a bit ?
I would say though from experience reading on here about other countries, there are so few specialists that they are all getting lots of referrals and waiting times are getting longer & longer, even in the US which used to be quick, so as it’s only been a couple of months for you it’s maybe not that long in the scheme of things 
It might be possible to try some of the blood thinner medications or supplements which members have tried to help a bit with your symptoms while you wait, there’s info here:
ES Information- Treatment: Pain Relief, Medications and Alternative Therapies - Welcome / Newbies Guide to Eagle Syndrome - Living with Eagle
Sending you a hug 
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