I agree with the above post. It was not uncommon to find British units and American units on the same battlefield, fighting right next to each other, with the Brits outfitted in American Sherman tanks. The military command structure for Britain and the US was unified under Dwight Eisenhower, so they pooled manpower, and the US war industries kept everyone supplied with materiel.
There was also cooperation in naval operations, convoy operations and escort duty between the British and Americans to deliver those supplies to ports in Britain and later France and the Soviet Union. Medical units also cooperated, with American and British doctors treating their own soldiers and allied counterparts alike. Cooperation with the Soviet Union was pretty much limited to supplies, mostly trucks. So other than making the Red Army a little more mobile, they pretty much fought on their own outside of some general agreements on strategy.