Several reasons have been postulated, e.g., scenes of Édith's activities during the war were shot but ended up on the cutting room floor, the director was afraid he couldn't find a balance between too much and too little about WWII so dodged it entirely, it might have been too much of a distraction from the fact that the movie was not about the war but about Édith Piaf, the French may have thought of her as a traitor for entertaining Germans, film length, cost effectiveness, the fact that not much is known about Édith's activities during the war, etc. and etc.
What IS known about Édith's activities during WWII is that she was considered to be a marraine de guerre (lit: godmother of war), a person who "adopts" soldiers and sends them letters, cigarettes, and food parcels. Because of this, she was welcomed as an entertainer of French prisoners in German stalags. In doing so, she was able to bring them cigarettes and to smuggle in a few other things, like compasses and forged sets of identity papers, to help them escape. In a short biography of Édith Piaf written by Christie Laume, (the sister of Édith's second husband, Théo Sarapo), Laume tells how Édith posed for pictures with French prisoners. When Édith returned to France, she made individual passports for the prisoners, using the pictures taken in Germany during her visit. When she went back to sing for them, Édith gave each one a passport that brought them to freedom. [See: http://www.christielaume.com/biography_edith.html]
In her autobiography, Édith describes how she and Sacha Guitry put on a show to raise money for the families of 50 French soldiers who had been recently killed when their stalag was bombed. In the Wikipedia entry for Édith Piaf, citations are provided that tell how Édith gave the Jewish composer of L'Accordéoniste, Michel Emer né Michel Rosenstein, sufficient money to enable him to hide during the war. It also describes how Édith would go into concentration camps with her small band of accompanists and come out with a dozen or more, claiming them as members of her troupe. In this way, she is said to have saved over 100 people, until the Nazis realized that she came out of the camps with more accompanists than she went in, and they stopped inviting her.