It is impossible to know how the war would have turned out with the help of animals.  However, the war would have definitely been fought differently.  Without pigeons, dogs, and cats to deliver messages, it would incredibly difficult to get information past difficult terrain.  Without the mules and donkeys carrying heavy loads, the governments of various countries would have either spent way more money getting vehicles or would have been unable to have certain supplies.  The use of animals in World War II also led to future important scientific research.  For example, while "M-dog" training was eventually abandoned, their training paved the way for modern methods of using dogs to detect explosives, which saves the lives of soldiers in warzones.