The League of Nations throughout the 1930s had attempted to stop the build-up that eventually led to World War 2. However, virtually all Axis members had left it by 1937 (there was still Finland and Thailand, but both joined the Axis out of desperation), and the fundamental structure of the League of Nations made it impossible for considerable diplomatic action to be taken against Italian, German, and Japanese aggression.

At the outbreak of the Second World War, with the German or even French invasion of Switzerland possible, a lot of the powers were transferred to the General Secretary Joseph Avenol on Sept. 30, 1938 and on Dec. 14 1939 to allow the League to operate in reduced capacity, and allow for decisions to be passed more promptly (ie: the Soviet Union being kicked from membership after invading Karelia). Because of this consolidation, the staff of other countries became meaningless, and almost all were recalled. The buildings in Geneva had sat empty from 1940 to 1946, and were soon transferred to the United Nations after the war.

The Basic run-down: The League of Nations was all but dismantled after the start of World War 2 with the general secretary being given considerable power, and any possible decisions could have been handled by governments-in-exile, the majority of which were in the UK. The shaky state of Europe and the fact Germany surrounded Switzerland made continued operations at the Palace of Nations in Geneva a massive risk, so operations primarily continued in London.