Have many other US words found their way into mainstream Australian English?
In the Gold Rush era, various terms such as the verb “prospect”, “digger” meaning a miner, and “bush” meaning a rural area are thought to have been transferred from American English to Australian English. Of these, “The Bush” is considered an Australianism. Similarly, “bonzer” is considered a archetypical Australian word, and the Macquarie dictionary reckons it derived from “bonanza”, brought to Australia by Americans.
According to Phillip Bell and Roger Bell in Americanization and Australia, “hitched” meaning married, “tough luck”, and “chip in” meaning to contribute were all listed as having been introduced to Australian English by 1936.
“OK” began to be used in Australia in the 1940s: WW2 represented a major opportunity for transfer as thousands of US soldiers were stationed in Australia.
Since WW2 of course there would have been countless examples of words that started in the USA that have become part of Global English and hence of course are used in Australia.
EDIT: one regrettable borrowing was the racist term “coon”
EDIT: The Australian Labour Party changed its name to the Australian Labor Party due largely to the influence of the American born labour leader King O’Malley. The spelling with the -u- was the standard in Australia at the time and has remained so since, but the u-less spelling for the ALP has stuck.