United State, Japan, and Russia all tried and eventually developed the atomic bomb in the 40's and 50's. Where did the idea originate? Where did the idea of the "atom bomb" come from and how did 3 countries come to the idea only in conditions of war?

There's a little confusion here. (Japan didn't develop atomic bombs, for one.)

But if you're asking, where did the idea of nuclear weapons come from, see this post, which addresses the scientific and political preconditions for people all over the world to start thinking about the feasibility of atomic weapons. Many countries had small investigatory research programs; only the United States actually launched a full development program during World War II. The Soviet Union launched its own development program after World War II, in response to the United States.