Ocker is in the OED (The complete edition), with references to the television sketches, the name Oscar, and the definition as a rough, uncultivated Australian. Recent editions of the OED have added definitions from the Australian language. My latest is 1987; they hadn't caught up with Oz at that time.
Aussie showed up in print in 1917 as Ausie, and in 1918, as Aussie in "The Australian Soldiers' Magazine." Looks like it originated with the soldiers (quotations given in the OED).

Ran across an old word, "australize," to point southward. 17th century, but no one had a real need for it, so it quickly became obsolete (if it ever had common currency). Perhaps it could be resurrected to point to attempts to integrate immigrants (but already the term ockerize is available).