Nuclear forces work for the US Strategic Command, who report to the SECDEF who reports to POTUS.
POTUS SECDEF > STRATCOM > Assigned nuclear forces.
Nuclear Command and Control and Communications (NC3), is managed by the Military Departments, nuclear force commanders, and the defense agencies. NCCS facilities include the fixed National Military Command Center (NMCC), the Global Operation Center (GOC), the airborne E-4B National Airborne Operations Center (NAOC), and the E-6B Take Charge and Move Out (TACAMO)/Airborne Command Post (Looking Glass)
Decisions on the employment of nuclear weapons can be made only by the National Command Authority (NCA), which consists of the President of the United States and the Secretary of Defense, acting in concert.
https://en.m.wikipedia.org/wiki/Nuclear_command_and_control
https://en.m.wikipedia.org/wiki/National_Command_Authority_(United_States)
This would confirm that the CJCS is not in the nuclear chain of command, but does advise the POTUS based on what is reported on the MSM or whatever.
Apparently on 9/11, STRATCOM made their response decisions based on the watching CNN. Interestingly, they were on the final day of a big wargames exercise. They launched the E4-B "Doomsday" plane assigned to the 55th Wing fearing a bigger attack.
The current commander of STRATCOM is Charles Anthony "Chas" Richard (born 1959)[1] is a United States Navy admiral who currently serves as the 11th commander of United States Strategic Command. Former commander of Submarine Squadron 17, Richard is a native of Decatur, Alabama, and is a 1982 graduate with honors from the University of Alabama. He has earned master's degrees with honors from the Catholic University of America and the Naval War College.
https://en.m.wikipedia.org/wiki/Charles_A._Richard