I doubt the military would want an ethical machine in control of their weapons. The probability of it deciding not to launch when ordered to on ethical grounds would be far too high. Respect for life and the responsibility to vaporize millions without hesitation when ordered to do so don't mix very well.
That is assuming that humans would no longer have control over their infrastructure. Any AI is going to have some default programming that implies not hurting anyone, and taking a backseat to human command. Skynet was originally established as a method of defense, not of aggressive warfare. That's just what it became accidentally.
It's a lot easier to believe in an ethical computer becoming a government program trusted to run everything, than an emotional child-like computer becoming the same program.
The point of automating nuclear missiles is defense through MAD. You need a system that will launch everything even if the human leadership and infrastructure is destroyed. To paraphrase Dr. Strangelove, A doomsday weapon isn't much of a deterrent if there is a way to stop it.
What they'd want is an AI that obeys the rules slavishly "If Russia attacks, launch all of the nukes at their predefined targets". It's the simplest and best defense. The Russians won't attack if they know that it is impossible to avoid being nuked in retaliation.