Short history essay coming. I've long been paid for writing this kind of thing, so plunging into it is a kind of occupational hazard. Avoid if you like.
Commonwealth countries were viewed after WWI as central to any future British war effort - something London had to worry about during the late 30s. One of the reason that Chamberlain had such broad support for "appeasement" was that it was wildly popular in the Commonwealth countries. In all of them there was talk, often heard in the isolationist circles in the USA, warning their governments not to shed blood to save the British Empire. As it was Canada, Oz and NZ all declared war on Germany in 1939 with the UK. But they all supported the war effort with conditions. Full conscription for overseas service took place only in 1943-44 in Oz and Canada. There were, however, large numbers of volunteers. Oz insisted that their units be under Australian command, but allowed their people to directly join the RAF or RN. Bomber Command in particular was very dependent upon Commonwealth volunteers. (Should note that New Zealand was the most cooperative, probably because so many people were first or second generation immigrants. It was the only Commonwealth country that had strong supporters of a hard line policy against Hitler.) Pearl Harbor upset part of the applecart. Australia had raised four divisions of volunteers to serve overseas - Australian Imperial Forces (AIF). Three were serving in the Mideast. (6th, 7th, 9th AIF) Churchill lost his cookies and persuaded Canberra to send the fourth (8th) to Singapore at the last minute where it was lost in the debacle. Oz very publically seperated itself from London and appealed to the US for support. They brought home the 6th and 7th AIF immediately much to Churchill's dismay. They asked for Douglas MacArthur to serve as CinC figuring (correctly) that his high standing in the US would draw US forces to Oz. (Not many - the SW Pacific was not much higher up the strategic totem pole than CIB. MacArthur bowed to necessity and specialized in economy of force until Leyte.) Only with reluctance did Oz PM Curtin allow Montgomery to borrow the 9th AIF for Alamein - the price was the allocation to Oz of the badly crippled 1st Marine Division from Guadalcanal in December. (You can bet Montgomery would have cherished an Australian corps for the campaign in France. The 9th AIF returned to Oz immediately after Alamein. However he had a Canadian equivalent.) The result of all of this was the completely unplanned allied offensive in the Southwest Pacific which ultimately split Japanese defenses in two and largely explained what was considered by Marshall and company to be impossible in early 1944: the defeat of Japan in 1945.
While all of this was going on Commonwealth volunteers (notably the Canadian Navy) were aiding British combat forces and helping to feed allied armies and Britain itself. They paid a price. Both Canada and Australia lost approximately 40,000 men KIA (50% or so of Australian battle casualties were in Europe: 7,000 perished in Japanese POW camps, the remainder of combat deaths came fighting the Japanese in New Guinea) and New Zealand suffered 11,000 killed, mostly in Europe. It's worth noting that the UK and Commwealth forces all participated in the Cold War. It's also worth pointing out that a major result of WWII was to accelerate the full psychological independence of the Commonwealth nations from Britain.
Eric