What year did the arms race start?
In 1939 when nuclear fission of uranium-235 was discovered and reported in the scientific journals Germany, England, France, and the USSR began a race to see who could produce an atomic bomb first. In 1940 Germany took over France and absorbed their atomic bomb project. In 1942 the US entered the war and England began moving their atomic bomb project to the US, fully merging it with the Manhattan Project in 1943. Germany's atomic bomb project was effectively shut down in 1943 with the remaining research limited to small heavy water moderated reactors that might be scaled up after Germany won the war to make power plant reactors. The USSR's atomic bomb project was severely resource limited and made no significant progress during the war (except to collect information on the British and US projects through spies).
However what is usually called the "nuclear arms race" actually started September 23, 1949 when president Truman announced to the world "We have evidence that within recent weeks an atomic explosion occurred in the U.S.S.R.". This test explosion was on August 29, 1949 at 7:00 AM local time.