No, they weren't "ordered into Shroomanian waters to fire off the nukes". What happened was, and I quote:
Coiler wrote:"Yes. I sent the submarine, knowing full well that it would collide."
That's of course Marina talking. And there's nothing about a nuclear launch there. In fact, according to
Coiler the nuclear war between Coilerburg and Shroomania is still
two years into the future. There's no way the captain would've known about it, let alone know of any order to fire.
Now, I personally feel we'd be better off retconning Marina out of it, and having the submarine be where it was because President Farve wanted to see how easy he could get his subs into Shroomanian waters, as a prelude to the invasion of that island chain he was planning. In fact I've written most of my posts assuming that's what happened, with Farve being deposed the minute people found out what he was planning.
The point remains however that under neither scenario would there be orders to kickstart the nuclear armageddon aboard that submarine, or in the captain's head, or anywhere else.