Trump has the final say. No one actually has the ability to stop him if he wants to drop nukes. His cabinet can try to suggest to him that would be a bad idea, but it's important to remember that our president is obviously suffering from serious mental deterioration.

Every speech he gives sounds half the same. He can't keep a coherent train of thought while speaking. Watch any video of him from 15-20 years ago and is clear that his brain power has since dropped greatly. This seems to make him open to suggestion (he seems to always agree with whoever spoke to him last), but he has often changed his opinion on a dime. What's to say he doesn't get pissed off enough by NK's taunting he decides to just nuke them without considering the consequences.

Many in his base seem to support striking NK pre-emptively, thinking it will prevent a Pearl Harbor type attack. All he really cares about is appeasing those people. He has tried to please them as best as he can, but for the most part, he holds no sway over legislators (can't repeal and replace). He has tried to use executive orders (Muslim ban), but his more controversial ones have been declared unconstitutional. But he has the full authority to order nukes since our constitution still needs to be updated for the 19th century, so it's much to vague regarding the president's unilateral abilities to control the weapon advancement of the following century.