Defense Primer: Command and Control of Nuclear Forces [Updated September 30, 2021] [open pdf - 536KB]
From the Document: "The U.S. President has sole authority to authorize the use of U.S. nuclear weapons. [...] The President can seek counsel from his military advisors; those advisors are then required to transmit and implement the orders authorizing nuclear use. [...] General Milley, the current Commander of the Joint Chiefs of Staff (CJCS) [...] noted that he is a part of the 'chain of communication,' in his role as the President's primary military advisor, but he is not in the 'chain of command' for authorizing a nuclear launch. He also noted that, if the President ordered a launch, the CJCS would participate in a 'decision conference' to authenticate the presidential orders and to ensure that the President was 'fully informed' about the implications of the launch. The President, however, does not need the concurrence of either his military advisors or the U.S. Congress to order the launch of nuclear weapons. In addition, neither the military nor Congress can overrule these orders. As former STRATCOM [U.S. Strategic Command] Commander General Robert Kehler has noted, members of the military are bound by the Uniform Code of Military Justice 'to follow orders provided they are legal and have come from competent authority.' But questions about the legality of the order--whether it is consistent with the requirements, under the laws of armed conflict (LOAC), for necessity, proportionality, and distinction--are more likely to lead to consultations and changes in the President's order than to a refusal by the military to execute the order."
CRS In Focus, IF10521
Congressional Research Service: https://crsreports.congress.gov/