Jews arrived in Australia on the very first day of European settlement - 26 January 1788, when the First Fleet sailed into Botany Bay with between six and fourteen Jews among the convicts being transported from England to the new penal colony on the other side of the world. One of them, Esther Abrahams, transported for stealing a small quantity of black lace from a London draper's shop, married George Johnstone, a Royal Marine officer she met en voyage. He ultimately became Governor of New South Wales, and so Esther the convict lass would up as First Lady of the colony. Although her children with Johnstone were brought up as Christians, her descendents were not entirely lost to Judaism - some years ago, a woman who traced her descent to Esther entirely through the female line, and is thus Jewish according to Halakhah, reconnected with Judaism.
Esther was by no means the only Jewish convict who made good. Many became successful in both mercantile and public life. During the nineteenth century, especially after "free" (as opposed to convict) immigration began in earnest in the 1820s, there were many Jews in country towns in Australia. A number of Jews, not only from England but from continental Europe too, kept hotels and acted as storekeepers during the gold rushes of the 1850s. Unless their descendents moved to one of the big cities, where Jewish communal life was more viable and there were more prospective marriage partners, the rural Jewish families tended to become totally assimilated, with the result that around 250,000 Aussies today can claim a Jewish ancestor.
During the inter-war period, when, incidentally, Australia had a Jewish Governor-General, Sir Isaac Isaacs, as well as an iconic Jewish military hero, Sir John Monash, the Australian Jewish community seemed on the verge of disappearance owing to the inroads of apathy and assimilation. The arrival of thousands of Jews from Eastern Europe both immediately prior to and after the Second World War changed all that, and today Australian Jewry remains one of the most flourishing in the Diaspora. It is strongly committed to Israel.
Below is a comparable video of Jews in New Zealand, which, by the way, was never a penal settlement: