The League of Nations really was not related to the Cold War in any important way. It was no longer in existence by the time the Cold War really got started.
The League was created after WWI to try to keep another war from happening. The Soviet Union was part of the League and had a permanent seat on its council. The US was not part of the League. The League started to fall apart before WWII because it was unable to prevent countries from invading one another (Italy invaded Ethiopia, Japan invaded China). It clearly had no real power.
After WWII, the League officially disbanded. Its organization and much of its property, however, became the basis for the United Nations. The United Nations was involved in the Cold War, trying to keep the peace and serving as a forum in which the two sides could interact relatively peacefully.
Since the League of Nations dissolved in 1946, it is hard to say that it was related to the Cold War. The only real connection is that it transformed into the United Nations, which was involved in the Cold War.