There actually was a fair amount of migration from Australia to the United States, beginning with the California Gold Rush of 1848-55. Given that there was no transcontinental rail system (nor a Panama Canal) at this time, it was actually easier to reach California from Australia than it was from the eastern seaboard of the United States. At the same time, Australia was facing an economic slump. So thousands of Australians and Kiwis in 1849-50 decided it was a better bet to spend ten pounds for a trip to San Francisco than to look for work at home.

Meanwhile, a British prospector, Edward Hammond Hargraves, who had tried and failed in California, sailed to Australia in 1850 with the hope of finding gold there. Seeking terrain similar to that of the California gold fields, he actually succeeded in discovering a goldfield in New South Wales in 1851 - touching off an Australian gold rush.

For the remainder of the 19th century, there developed an international subculture of prospectors who chased discoveries of gold and silver wherever they took them: Australia, New Zealand, British Columbia, Nevada, Colorado, Idaho, Transvaal, and finally the Klondike. Australians and New Zealanders were part of this miners' culture, which would have explained their presence in the United States in the 19th century.

So while an Australian or a New Zealander would have been an oddity in Wyoming in the 1870s, it's not impossible that such people did exist.

Sources: Jay Monaghan, Australians and the Gold Rush: California and Down Under, 1848-1854; H.W. Brands, The Age of Gold: The California Gold Rush and the New American Dream