First, you need to get your ass out of Europe. I have no idea how you'd do that without getting a stamp saying how long you've been there, but to apply for UK residency, you cannot be in the UK, and I'm pretty sure you won't be able to apply if you're illegally in the EU.
Now, once you're back in Australia, there are a couple of ways you can go about this. This is all information taken from the UK Border Agency website, so I'd suggest you check that out.
If you are not a British citizen, you may still have the right of abode if, on 31 December 1982, you were:
a Commonwealth citizen with a parent who, at the time of your birth or legal adoption, was a citizen of the United Kingdom and Colonies because he/she was born in the United Kingdom
If you're not that old, easiest way is through ancestry (though, this is JUST for the right/visa to live and work there... not residency):
You can come here to work if you can show that:
you are a Commonwealth citizen;
you have a visa or other form of permission to enter the United Kingdom because of United Kingdom ancestry, or your permission to stay here in this category has previously been extended;
you are aged 17 or over;
you have a grandparent who was born in the United Kingdom (including the Channel Islands and the Isle of Man), or a grandparent who was born before 31 March 1922 in what is now the Republic of Ireland;
you are able to work and you plan to work in the United Kingdom; and
you can adequately support and accommodate yourself and your dependants without help from public funds (see rights and responsibilities for an explanation of this).
From what I can tell, after five years of being there legally, you can fill out Form SET(O) claiming UK Ancestry for residency (indefinite leave to remain).
In terms of which forms to fill out, I would check visa4UK :: Welcome
and UK Border Agency | Home Page
General disclaimer: I am not an expert on how to get your UK residency and I might have misinterpreted something on the website, so double check everything I said and do not sue me if I was wrong :P
Hope this helps, though.
Brazil, Canada, Greece, Turkey, Mexico, England, Wales, Ireland, Scotland, Northern Ireland, Portugal, The Netherlands, USA
Belfast, Northern Ireland