When did League of Nations become the United Nations?

The League of Nations never "became" the United Nations. The League of Nations was formed as a result of World War 1 and fell apart as the world moved into World War 2. During the War, those national with which the US and Britain were allied were often called "The United Nations." After the War, in 1946, a formal organization called The United Nations was created, but it was not a continuation of the old League of Nations, which had been discredited by its inability to respond to the increasing belligerency of Germany and Italy.