If a Nuclear bomb were dropped....

[Excited Penguin]

When you're talking about a nuclear bomb you're talking about certain governments by necessity. A random group of people with no government resources behind them won't be able to pull something like that in a million years. Nuking those governments and/or the sites that hold further nuclear weapons would be the only thing left to do.

Don't be so sure. The amount of fissile materièl available is plenty:

There have been no confirmed reports of missing or stolen former-Soviet nuclear weapons, but there is ample evidence of a significant black market in nuclear materials. The International Atomic Energy Agency (IAEA) has reported more than a hundred nuclear smuggling incidents since 1993, eighteen of which involved highly enriched uranium, the key ingredient in an atomic bomb and the most dangerous product on the nuclear black market.

http://www.cfr.org/weapons-of-mass-destr...ukes/p9549

The contradiction in your post won't be addressed.

I guess my question is: How much power would the POTUS really have in that kind of situation? Could he just proceed without the backing of Congress, for example?

In that sort of instance, he would have the backing of Congress, no questions asked. Fear is not only the mind-killer, it is the mind-seller.

It's not so much a contradiction as it is bad phrasing. I was continuing an earlier thought and didn't separate it from the random terrorist group line of thought. My bad.

The fact that there's a black market for certain materials doesn't mean it's not governments who are buying them. It's still quite a stretch to think a random group of people could both acquire and/or make atomic weaponry on their own.