Nato troops serve their individual countries.
American troops on NATO exercises are American troops.
When in combat there are command decisions that means someone else can command troops.
In WW2 Eisenhower was supreme allied commander which meant British and Canadian troops were commanded by him, but they stayed British and Canadian.
During the Battle of the Bulge, Montgomery, a British general, was given temporary command of all allied troops, including Americans, North of the bulging line.