As to the state of rapport between the British and American commands in WWII, I would refer you to the movie "Patton". I am assured by several senior members of our local American Legion Post that the portrayal in this movie is entirely accurate. The supreme Allied Commander was an American (Eisenhower) and the Supreme NATO Commander since the war has always been an American General.
This does not mean that there was no cooperation in specific cases, or out of immediate necessity, but the ultimate authority has always been the American Chain of Command. (At least until Bubba got hold of it!)
On May 3, 1994, President Clinton signed Presidential Decision Directive 25 (PDD-25), a policy directive outlining the administration's position on reforming multilateral peace operations. PDD-25 defined requirements that must be satisfied before the U.S. will participate in international peacekeeping operations. In one of PDD-25's provisions, the Clinton Administration clarified the position of the United States with regarding command and control of United States military personnel participating in a multilateral peacekeeping operations. The Directive defines "command" of United States armed forces and "operational control" of those forces, distinguishes between the two, and maintains that the President never relinquishes "command" over United States military personnel, even though he he may place United States military personnel under the "operational control" of a non-U.S. commander for limited and defined purposes. If the U.S. relinquished "command" over those forces then they wouldn't be able to recall them or replace them with other units. But that didn't happen.
Given those parameters then the U.S. frequently placed whole armies under foreign operational control during World War II, while maintaining overall command over those forces.