The bombs shown in the sequence following the Berlin air lift are not early American nuclear bombs of the 1950s but conventional British "Tallboy" or "Grand Slam" bombs from the Second World War. They were very large and heavy but filled with conventional chemical explosives and thus had only a fraction of the power of actual atomic bombs.
The footage shown with the caption "United Kingdom Tests A-Bomb" does not show the first British nuclear test ("Operation Hurricane") which had a completely different, very distinct mushroom cloud. Rather, the footage depicts an American test shot in Nevada.
The footage shown when the narration mentions American Jupiter missiles stationed in Turkey in the 1950s actually shows a German V2 rocket from WWII, towed behind a German Sd.Kfz. 7 half-track.
When the narration describes the conversion of the strategic bomber forces to jet planes such as the B-52 in the 1960s, the footage does not show a Soviet jet bomber from that era but a Tupolev Tu-160 which was only introduced into active service in 1987. Its predecessors from the 1960s, the Tupolev TU-16 and Myasishchev M-4, never achieved the range of a truly strategic intercontinental bomber. Instead, the Soviet Union had to keep relying on propeller-driven Tupolev Tu-95s up until the late 1980s and Russia keeps these aircraft in service to this very day, including the war against Ukraine.
When the narration mentions the Soviet series of large hydrogen bomb tests from the 1960s that included the biggest nuclear device ever tested, the "Tsar Bomba" with a yield of 58 megatons, the footage does not show a Soviet hydrogen bomb test at all but rather the Baker shot of Operation Crossroads from 1946, with a yield of only 18 kt, just a tiny fraction of a large Soviet hydrogen weapon's power.