How did the role of the Canadian government change over the course of World War II?

Canada officially declared war on Germany just over one week after World War II officially began on September 1st 1939. This was Canada's first independent declaration of war. 

Initially Canada was reluctant to join the war or spend money on the armed forces, but managed to garner impressive participation, with 10% of their population joining the armed forces. But they still entered the war with a moderate war strategy, stating that it would prioritize Canadian defense, possibly assist other countries, and serve allies primarily by providing food and supplies. By the end of World War II; however, Canada's military was one of the largest in the world. 

During the first nine months of the war, Britain was essentially responsible for Canada's war plan. They trained British pilots and their equipment was made according to British designs. However, during the Battle of France in 1940 Britain began to run out of equipment and soon thereafter told Canada they would need them to provide more. Many piloting graduates who were expected to be trainers were sent to Europe to fight as needed and Canada's involvement increased. They also began to send troops to defend the West Indies.

After France fell and the chance of Germany invading the Americas became a real possibility, Canada began to shift its focus from aiding Britain to defending itself, sending troops to their eastern colony of Newfoundland. Toward the end of 1940 a defeat of Britain by Germany became increasingly likely, so much so that it was agreed that the United States would overtake control of Canada's military should Europe be overtaken by Germany. In the Fall of 1941, Canada offered troops to the British government to help defend Hong Kong. They began to fight Japan there and eventually the allies lost. 

Canada was made responsible for two very strategic points in the Atlantic ocean during World War II: the Mid-Atlantic Gap, and the English Channel. The former, near Greenland, was a very hostile point in the supply chain and the gap was eventually closed by the Canadian Navy in 1943. The latter they controlled during the invasion of Normandy. The U.S. and Britain relied on Canada to cover their flanks during the invasion.

The Canadian government increasingly pressured Canadian troops to be put into action, and as a result they participated in the Dieppe Raid on occupied France, which, after the death and capture of over half the troops sent, was largely considered useless. Public pressure also pushed the Canadian troops to participate in the invasion of Italy in Sicily.

On June 6, 1944 the Canadian forces joined the Battle of Normandy. They experienced significant losses at first but eventually penetrated farther than the United States or Britain. 

In the end, the growth of the Canadian military throughout the time of World War II, from a small supportive military without a lot of funding to a world superpower, was impressive. By the end of the war, their air force was the 4th-largest in the world and their navy the 5th.