On May 10, 1940, Germany began invading the nations to its west. The advanced and overpowering German army cut through France using blitzkrieg tactics. In six weeks, France was defeated, and Germany had gained control of France. The result was also the collapse of the French government.

At the outbreak of the war, the French leader was Paul Reynaud. Reynaud would resign on June 16, 1940, and would be replaced by Marshal Philippe Pétain. Technically it was Pétain who was running the government in France, known as the Vichy government based on its capital. Vichy France worked closely with the Nazis in an agreement to avoid further conflict with them.

There was, however, also a government in exile. French military leader, Charles de Gaulle, led the government in exile from Britain during the war. Charles de Gaulle encouraged his countrymen back in France to resist the German occupation. De Gaulle would also become leader of France's provisional government as France was liberated from Nazi occupation in 1944.

This is a somewhat complicated question as there were a number of different people who could be called the leader of France during the war.

When the war began, the leader of France was Edouard Daladier.  He was the prime minister when war broke out in 1939.  Of course, France was defeated rather quickly in the war.  When this happened, part of France was occupied by Germany but the other part, known as Vichy France, retained nominal independence.  The leader of Vichy France for as long as this entity existed was Philippe Petain.  At the same time, however, Charles de Gaulle was seen as the leader of the Free French in exile.  

All three of these men, then, could be called the leader of France during WWII.