It's pretty much impossible to separate the British colonization of Australia, and later New Zealand, from the ARW. James Cook claiming the eastern coast in 1770 is hardly relevant, everybody and their grandma had claimed the entirety or part of Australia by that point. And just like with all the other countries that claim didn't actually originate from or result in an interest to settle the place.
For starters the British decision to settle Australia in 1786 was the end result of over a decade of negotiations in the British government about the creation of new penal colonies. Those negotiations were themselves caused by the outbreak of the ARW, and Australia wasn't actually considered as an option until after that war had already ended. Until the winter of '84-'85 the only thing being seriously considered (and tested) were various locations in Africa. Secondly the reason why near the end of the negotiations Australia actually ended up on the list of possibilities (and was eventually chosen) was because of the American loyalist James Matra. James had the idea to resettle other American loyalists on Australia, managed to present this idea to the secretary of state himself, and got an amended version of his plan (now including convicts as well) delivered to the prime minister in late 1784.