Interesting that the movie never mentioned which country nuked the U.S.........Lithuania?
In the novel “On The Beach” with a similar theme, WWIII begins in 1963/64 with a nuclear attack by Albania on Italy, and then escalated with the bombing of the United States and the United Kingdom by Egypt. Because the aircraft used in these attacks were obtained from the Soviet Union, the Soviets were mistakenly blamed, triggering a retaliatory strike on the Soviet Union by NATO.
There is also an attack by the Soviets on the People's Republic of China, which may have been a response to a Chinese attack aimed at occupying Soviet industrial areas near the Chinese border. Most, if not all, of the bombs included cobalt to enhance their radioactive properties.
In the movie version, unlike the novel, no one is assigned blame for starting the war, which attributes global annihilation to fear compounded by accident or misjudgment.