The bombings of Hiroshima and Nagasaki took place on August 6th, 1945 and August 9th, 1945. This was the first time atomic weapons had been used in war. The bombs we're so devastating that they wiped out enitre cities and also killed thousands of people. The horrific event left around 70,000 individuals in Hiroshima dead, and an estimated 40,000 dead in Nagasaki. The death that lingered after the bombs was prevelant as many people started getting radiation poisoning and dying almost immediately. Those who did not die suffered from burns, injuries, sickness, and deformation due to the high levels of ionizing radiation. After Hiroshima and Nagasaki were bombed, many individuals decided to leave and start new lives elsewhere. Ishiguro's parents, Shizuko and Shizuo Ishiguro, were both in Nagasaki at the time when the atomic bomb dropped and little is known about the rest of his family. Ishiguro, who was born on November 8th, 1954, left Japan at the age of two with his mother, sister and father.

Overall, while there aren't many references to war in Never Let Me Go, Ishiguro has done multiple other books about life in Japan after the bombings of Hiroshima and Nagasaki. These books include An Artist of the Floating world, When We Were Orphans, and A Pale View of Hills. While this doesn't specifically show up in Never Let Me Go, deeper research on the life and works of Ishiguro reflect his attention to detail and subtle elements of fate and belonging in this work. 

Sources:

Britannica: Atomic Bombings of Hiroshima and Nagasaki

Life of Ishiguro

Aftermath of Bombings of Nagasaki and Hiroshima

(Previous Knowledge about WWII from High School History Classes).