It was in 1939. Einstein was already living in New Jersey. It is an interesting fact that although Einsteins signature was on the first letter Roosevelt received warning of the possibility of the Germans developing a fission bomb, it was really Leo Szilard who drafted the letter and got Einstein to sign it.
Einstein hadn't considered that a fission chain reaction might be used to build a bomb until Leo Szilard brought the subject up. Leo Szilard, knew Einstein from sometime in the 1920s when they were both living in Berlin. Szilard is credited with independently realizing that Fission could be used to create a chain reaction. Szilard always claimed the idea came to him while waiting at a traffic light in London. Fission itself had only recently been discovered in Germany by Otto Hahn and Strassmann . But it was the Scandinavian Lisa Meitner who correctly explained the result of their experiment.
Szilard went to Einstein and explained to him that fission could possibly be used to create a chain reaction releasing a great quantity of Energy. At that time uranium was coming from the Belgian Congo. Szilard had remembered that Einstein was a friend of Belgian Queen Elizabeth. Szilard wanted to warn her that the Nazis might try to buy Congo uranium, and thought Einstein might be helpful in getting through to her. At some point Szilard roped in Teller. They warned the Belgian Ministers that Germany might attempt to obtain Uranium from the Congo. At the time, Szilard himself was working on the concept of a chain reactions at Columbia University. Szilard thought Einstein, because he was famous, whereas Szilard was not, might be able to use his influence to get word to Roosevelt of the possibility of the Germans developing a fission bomb . They first tried to go through Lindbergh, but that was unsuccessful. (They weren't aware that Lindberg had just received the Medal of Honor from the Nazis.) Then they hit on the scheme of using another Hungarian Szilard knew, Alexander Sachs, to get through to Roosevelt -- Sachs was a Lehman Bros Economist and was a friend of Roosevelt. In August 1939, Szilard drafted a letter that he got Einstein to sign. They gave the letter to Sachs. Sachs didn't deliver the letter until the second week of October. Roosevelt responded with a thank you note to Einstein, but very little was done until Szilard got Einstein to sign another letter addressed, this time, to Sachs. Sachs took the letter to Roosevelt and read it to him. This time, March 1940, and the President pushed harder to get the matter looked into with urgency.
So we have Leo Szilard to thank for setting the U.S. on a course to develop the bomb, but Einstein's signature on the letters sent first to Roosevelt and then to Sachs were important in getting the attention of the right people needed to get the project started.