This is Tyler Cowen's excerpt of this NYT story:

…there were at least two instances in which top officials tried to slow, or undermine, the president’s nuclear authority.

The first came in October 1969, when the president ordered Melvin R. Laird, his secretary of defense, to put American nuclear forces on high alert to scare Moscow into thinking the United States might use nuclear arms against the North Vietnamese.

Scott D. Sagan, a nuclear expert at Stanford University and the author of ‘The Limits of Safety,’ a study of nuclear accidents, said Mr. Laird tried to ignore the order by giving excuses about exercises and readiness, hoping that the president who sometimes embraced the ‘madman theory’ — let the world think that you are willing to use a weapon — would forget about his order.

But Nixon persisted. Dr. Sagan reports that during the operation, code-named Giant Lance, one of the B-52 bombers carrying thermonuclear arms came dangerously close to having an accident.

Then, in 1974, in the last days of the Watergate scandal, Mr. Nixon was drinking heavily and his aides saw what they feared was a growing emotional instability. His new secretary of defense, James R. Schlesinger, himself a hawkish Cold Warrior, instructed the military to divert any emergency orders — especially one involving nuclear weapons — to him or the secretary of state, Henry A. Kissinger.

It was a completely extralegal order, perhaps mutinous. But no one questioned it.

So the President's power when it comes to nuclear can be quite limited.