It being a global war, British troops are involved in theaters all across the world. British troops are present in Crete, trying to support the Greeks against the Germans; North Africa, where they fight both Italians and Germans until early 1943 while trying to protect Egypt and the Suez Canal; Italy, where the British jointly invade with the Americans in 1943; East Africa, where they invade Italian colonial possessions and Ethiopia; Singapore and Malaysia, where they are overrun by the Japanese in 1942; Burma, where the Japanese also successfully invade in 1942; and thereafter in campaigns to retake Burma in 1945. In short, British infantry fight in a whole host of different locations other than France in 1944, on top of the troops stationed in Britain for defense and thereafter in preparation for the invasion.

Now, important disclaimer: Britain also has an enormous empire that contributes manpower to the war effort. Canadians are present throughout the war in Europe, India (which raises a 2.5 million man army approximately, exclusively through volunteers; the largest volunteer army in history, if Srinath Raghavan's India's War is to be believed) is critical in East Africa and in Asia, particularly in stabilizing the British line following their various calamities in 1942, Australians and New Zealanders fight the Japanese, and Britain's African colonies contribute hundreds of thousands of volunteers. So not all of those troops I mentioned were exclusively Englishmen, but they do fight on behalf of the British government.

I understand the Navy and the RAF were involved in the battle of Britain, but more specifically, what would the infantry be doing.