On the Italian Front, the British Eighth Army and the US Fifth Army were always approximately the same size. In the summer of 1944, the British withdrew some of their forces for operations in Greece and the Aegean Sea and the US Army withdrew some of their forces for Operation Anvil Dragon, the invasion of Southern France. There were Polish, Indian, South African, New Zealand, Brazilian and Free French forces in Italy after August 1944, but the US Army was slightly larger than the British army in Italy in the later stages of that campaign.
On D-Day the British deployed three infantry, one airborne and the 79th armored division. The US Army deployed two and a half infantry and two airborne divisions. They reinforced the Normandy beachhead very quickly. Within four weeks, there were one million soldiers wedged into an area the size of Massachusetts. About sixty percent were US Army troops. By the fall of 1944, Montgomery's 21st Army group consisted of the First Canadian Army and the Second British Army and was only 40 percent the size of Omar Bradley's 12th Army Group, which consisted of the First, Third and Ninth US Armies. In France, I'd say the US Army became more numerous than the British army sometime near June 21st 1944.
In the air, the various US Army airforces began to outnumber the RAF in February 1944. This is counting all of the Tactical and Strategic air forces, based in both Italy and England. The Royal Navy was always larger than the US Navy in the Atlantic. Canada also contributed greatly to the Battle of the Atlantic. By August 1944, Canada had the third largest navy in the world and it did almost all of its fighting in the North Atlantic.
Sources: "Winged Victory: The US Army Air Force in World War II" by Geoffery Perret
"Order of Battle: Western Allied Forces of World War II" by Michael Haskew