The Lend Lease act allowed America to ship American arms to mostly Russia but also the other allies. Its been said that WW2 was won with American arms, Russian lives, and british intelligence. http://www.ourdocuments.gov/doc.php?flash=true&doc=71 The American and British forces cooperated closely on the ground during ww2 during the invasion of france and germany sometimes literally fighting together in battle. During D-day canadians americans and british all fought together.

In both the Great War and WWII, major allies (and smaller countries directly invaded, eg Belgium, Serbia, Greece) would have their own sections of the front, and fight more-or-less separate wars, with their own logistics trail. For example, the Western Front had a French sector, a British sector, and a (small) Belgian sector, and later on there was an American sector as well. This was a source of some friction between Americans and the others; the British and French would have liked to put in American divisions, or even battalions, to shore up their own fronts (incidentally putting the Americans under their own officers at the higher levels), but Pershing had instructions to fight as a separate army. This is why there were a lot of American units in France for a long time before they actually did any fighting; they were building up until they could go into action as an army.

Lesser allies' supporting troops would be slotted in wherever they fit, into the logistics and command organisation of a major; thus for example we find Greek battalions fighting as part of the British army in the African campaign, but external to the battalion they were just another Eighth Army unit - Montgomery or Auchinleck would say to their colonel, go, and he would go.