If a Nuclear bomb were dropped....

[Thumpalumpacus]

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.