Technically, everything there is correct: the unilateral authority for nuclear weapons use is in the hands of the president. In particular, the military are absolutely not allowed to do anything with nuclear weapons without the confirmation of the president. This system was put in place by Truman, and this decision probably saved the planet from nuclear war many times.
Now, in practice, the president, Trump or anybody else, cannot unilaterally start the nuclear war. He has to communicate with the military to activate the nuclear football and call NMCC. The order will be handled by EAC (emergency actions controller), who has to input their own codes, which needs to be confirmed by the superior officer. Choosing the nuclear option beyond what is already in nuclear football (where only retaliatory options are placed, not applicable in the first-use context) will require active input from STRATCOM, which will take hours. At all these points, there are many opportunities to intervene and/or seek additional confirmations from multiple authorities.
Tl;dr the nuclear _retaliation_ can indeed be launched by the president in minutes, if it is already proposed by the military and the only thing the president needs to to is to reply with nuclear launch codes from the biscuit and the football. Initiating a first use attack is a whole another story, will take hours, and most probably will not succeed at all.
P.S. Mark Milley is not in the nuclear launch command chain, so he has no authority to do anything about the procedure. Neither is Nancy Pelosi.