In the months after the destruction of Hiroshima and Nagasaki, President Harry Truman spoke of his ambition to make the atomic bomb “an overwhelming influence toward world peace.” Meanwhile, the Bulletin of the Atomic Scientists warned that “all we can gain in wealth, economic security, or improved health will be useless if our nation is to live in continuous dread of sudden annihilation.” Using specific events, people, and policies, analyze the impact of nuclear weapons on US foreign policy and the lives of Americans from the Manhattan Project through the Cuban Missile Crisis. How did different Americans perceive and make use of the bomb in debates about freedom and national security?

In the years that followed Hiroshima and Nagasaki, and certainly after the Soviets successfully tested the atomic bomb in 1948, annihilation by nuclear weapons was a constant fear for many Americans.
Atomic and nuclear weapons played a major role in foreign policy. Generally speaking, the United States spent billions expanding and testing its arsenal. The National Security Council memo NSC-68 advocated massive buildup of nuclear weapons in response to what it saw as the global threat posed by communism. This was seen as the only way to avoid war with the Soviets, a policy stance that would persist until the 1970s. The United States rapidly built up its capability, developing far more powerful weapons than the Hiroshima and Nagasaki bombs. On the other hand, no American leader was eager to use these weapons, and the fact that the Soviets also had nuclear capability was a major factor in promoting restraint in the escalating Cold War. In the Korean War, for example, President Harry Truman declined to use atomic bombs against Chinese troops and cities because he feared it would escalate a regional conflict into a global one. President Kennedy decided not to attack Soviet missile silos in Cuba (or to order a direct assault of the island) for similar reasons. Overall, the United States was committed to developing a massive nuclear arsenal even beyond the Cuban Missile Crisis.
Nuclear weapons also affected the everyday lives of ordinary Americans. The threat, imagined or otherwise, of nuclear destruction led to a sense of anxiety in the United States. Children regularly took part in air raid drills in public schools. Americans dug fallout shelters in their backyards, often at great expense. Many Americans, sensing that the stakes were raised, supported repressive measures against suspected communists: the HUAC proceedings and the McCarthy hearings were the most egregious examples of this phenomenon. On the other hand, the fear of nuclear attack fostered a renewed focus on education, especially in science and mathematics. The Soviet launch of Sputnik in 1957 caused many Americans—including policy makers—to conclude that the Americans had fallen behind the Soviets in terms of technology. Educating young scientists became a patriotic mission, one which led to the passage of legislation in the form of the National Defense Education Act that equated science education with national security.
All in all, "the bomb" was at the center of American foreign policy and the American consciousness for much of the Cold War era.
https://history.state.gov/milestones/1945-1952/NSC68

https://news.harvard.edu/gazette/story/2007/10/how-sputnik-changed-u-s-education/