Good information, thanks. My only quibble is that while the Australians formed a headquarters in the Middle East, it did not command the three divisions in action. The 6th Division fought in Greece and Crete as well as North Africa in 1940-1941, with mixed results. (If a Canadian division would have fought at that time, the results probably would have been the same.) The 7th Division served in the invasion of Syria, I believe, while the 9th fought in the desert in 1942. Note that these divisions were kept in the Middle East after the start of the Pacific War despite pleas from the Australian government. (The 8th Division was in Singapore and went into the bag, except for its commander, who decided he was too valuable to become a POW, and fled.)
Some source material I have read claims that the relationship between Chruchill and the Australian government was icy at best. There were a series of minority governments, characterized by confused and half-hearted support of the war effort. One of the PMs even conspired with Churchill's enemies in the UK with an aim of replacing him as the British leader, so Churchill's disgust with the Australians got personal. By the way, unlike Canada and South Africa, Australia had not invoked the 1931 Statutes of Westminster and therefore had its foreign affairs managed by Whitehall, so its negotiating position was weakened. Anyway, during the Japanese rampage across SE Asia and Oceania, the Brits retained the Australian army and ships of the RAN in the Mediterranean theatre. When they did return troops, at least one Australian brigade was sent to Ceylon. And, there were no modern fighter aircraft in Australia and the British dragged their heels about providing any or returning experienced RAAF pilots.
Sorry if this got off the thread, but the Australian experience at the time helps to explain their unstinting support of the Americans, who came to their rescue.