It probably worked differently over different periods of the Soviet Union (such is the experience of other nuclear states), but from what we can tell the way it was supposed to work in the 1950s-70s was that the Politburo could collectively agree that nuclear war was at hand and that this would let them authorize someone in the military to be the Supreme Commander who would have the authority to make any nuclear use decisions. How exactly what would work in practice is not clear. There are other indications that the Soviet nuclear forces were in practice controlled by the military and not really a distinct form of civilian rule as in many other nations.

Source: Pavel Podvig, Russian strategic nuclear forces (Cambridge, Mass.: MIT Press, 2001), 39-42 and Henry D. Sokolski and Bruno Tertrais, eds., Nuclear weapons security crises: What does history teach? (Carlisle, Penn.: Strategic Studies Institute and U.S. Army War College Press, 2013), 4-5. For a comparative look at how several states command and control policies evolved, see Salma Shaheen, Nuclear command and control norms: A comparative study (New York: Routledge, 2019).