I've been looking into this in a pretty dedicated way for a few months now, both as a matter of policy and a matter of history. You can find the results of my research on my blog in a three-part series here: Part I, Part II, Part III. There was also a Radiolab episode that I helped get made and an op-ed in the Washington Post that I wrote. My conclusions are based on my digging through obscure USAF doctrine, talking with a former Secretary of Defense, talking with former and current missileers, and doing a lot of historical research into the origins of our nuclear command and control system.

I only point out the sustained background research because there are a lot of people who, when confronted with something they think is illogical, will just dismiss it ("it can't be that way, that would be crazy"). I have been really shocked to have intelligent, educated, well-meaning people just outright reject the conclusions on this topic I've come to without any evidence on their side — simply because they don't like the answer, and don't want to believe it is true. It is a remarkable reaction, even by friends of mine who know that this is literally the kind of thing I study for a living.

The short answer is that everything I've been able to find points to the President having essentially unlimited and unique leeway to authorize nuclear strikes, with no legal mechanism by anyone else to veto the order, with no distinction between first or second strikes whatsoever. There are also fewer links in the chain of command than most people suppose (the Secretary of Defense is, I have come to believe, totally unnecessary for it, contrary to what a lot of people think); it goes pretty much directly from the President to the military (via the Joint Chiefs of Staff). There are good historical reasons for this: in the early atomic age, the question was seen about the President vs. the Military having this power (not the President vs., say, Congress or someone else), and the idea of having a deranged President was not really contemplated until the final days of the Nixon administration (but after that passed, the idea of possibly adding checks fell by the wayside again). The system is a Cold War system; it is about preserving deterrence, which is to say, it is about guaranteeing a rapid and overwhelming nuclear response. So everything is optimized for speed and efficiency, without time or methods for second-guessing.

One can postulate that in practice there may be some practical leeway involved — it's possible. But every rung of the system emphasizes that it is the President's authority alone that makes it happen. So I think it is rather optimistic to assume that there are secret checks on that. (Calling people "optimistic" on this matter is my favorite insult.) There are many, including myself, who think that this historical system is not a good idea anymore (no matter who the President is), but that's getting past the 20 year rule here. But my blog posts discuss some of the current policy issues regarding this, including a current bill before Congress (the Lieu-Markey bill), if you are curious.