Firstly, the rhotic R - the rolled R that distinguishes most American accents (but not the Boston accent, famously) - was not distinctively Irish in the late 18th century when Australia was originally settled. Much of England at this point would have used a rhotic R, in fact - and select English accents still do, though this has decreased in the mass media age. The part of England that had dropped the rhotic R at this point in time was generally centered around London and surrounds (as I discuss in more detail here). And by and large, the early convicts who came to Australia on the First, Second and Third fleets from 1788 onwards were generally in fact from South East England and very often London itself.

In general, Grace Karskens in The Colony argues that the Sydney colony was seen by the authorities as a way to reform people corrupted by city life by turning them into good productive farmers living honest rural lives - Sydney was not meant to become a city, but was meant to be a bare-minimum port for a spread-out rural settlement. As a result of this philosophy for Sydney, the convicts who were transported to Sydney were predominantly urban, and predominantly from London - very few of the initial convicts had any farming experience at all. Perhaps unsurprisingly, therefore, the London accent of the period is a prominent part of the Australian accent, which is why the modern cockney accent shares quite a few features with the Australian accent.

An observer of the Australian accent writing in 1820, James Dixon, argued that it seemed like a sort of smoothed out melange of English accents: "the children born in these colonies (in Sydney) and now grown up speak a better language, purer, more harmonious than...in most parts of England". Dixon here touches on something that is important in the creation of accents - it's children who create accents. Adults generally keep their accents, but children typically develop accents in tune with the population around them - and when that population is a mix of accents, children synthesise those into a distinct accent of their own. It's hard to know, of course, whether the Australian accent of 1820 was similar to the current one, or whether further immigration altered it, but an Australian accent already existed quite early on, as children were born in the colony and grew up to be adults - by 1820, there were plenty of (white) children whose parents had been born in Australia.

Irish emigration to Australia in large numbers occurred most prominently in the 1840s and 1850s - my Irish paternal ancestor who gave me my surname emigrated in 1842, for example. By 1842, the Australian accent had long been solidified - this was about 50 years after the colony was first established. Second generation immigrants typically speak with the accents of the area in which they grew up, rather than with the accents of their parents (with some minor variations); basically if you join a society before about puberty, it’s relatively easy to acquire the accent. So generally speaking, the children of my paternal ancestor from Ireland will have grown up to speak with the accent that the other children at school spoke with, which was likely some version of the Australian accent, with some influences from their parents.

In addition, Australia was part of the British empire, and for a long time has had a ‘cultural cringe’ about the worth of its own culture - including its accent - in comparison to that of the motherland. British educators coming to Australia such as William Churchill in the late 19th century sometimes expressed horror about the Australian accent and attempted to encourage Australian children to talk more properly (e.g., like English Received Pronunciation). The cultural cringe and the pressure to speak good English, especially after the rise of mass media such as radio broadcasting (which began in Australia in the 1920s), likely homogenised the Australian accent somewhat; it may have dampened some of the more obviously Irish influences on regional Australian accents.

One curiosity related to the Australian accent is related to a set of Australians who moved to Paraguay in 1893 with the intent to create a sort of socialist utopia; the surviving descendants who grew up speaking English have accents might therefore represent aspects of the late 19th century Australian accent that have disappeared in the modern Australian accent (or which might be explained by other cultural influences on the Paraguay settlement - typically the descendants of the New Australia settlers in Paraguay now learn Spanish in school and speak Guarani at home, according to research by Danae Perez). You can hear a third generation New Australian here.