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It all depends on how quickly the NK's can move south. If we can't move reinforcements over in time then things could get very nasty indeed. NK has been planning and refining it's options for decades and it's whole military revolves around the single purpose of a lightning invasion of the South.
I think it would be inevitable that a NK will have the upper hand in the early days of a war, but how quickly will SK/US turn the tide, that's the question.

how quickly will SK/US turn the tide

NK could take the whole penninsula. It's a possibility. Whether they could keep it is unknown. But they will probably be stopped in a couple of days and the fighting will continue for probably three weeks until NK is essentially gone as a fighting force. At that point China should come in, not to drive the UN forces out as was attempted 50 years ago, but to administer the country of NK until unification with SK.

"NK has been planning and refining its options for decades."

No doubt. But do you think the South Korean military as been playing tiddly-winks for those same decades?
The NK are not supermen. There may be a million of them or two, but they are NOT well-supplied with modern weapons, supplies, fuel, etc, and the DMZ is hardly an unguarded border.

People appear to have an elevated opinion of these guys due to the first Korean War, but the fact is that the North Koreans got their butts kicked all they way to the Yalu River, and it was the Chinese hordes who turned the tide of a quick UN victory into a bloody stalemate.

The nuclear wild card exists, of course, but the ROK military knows more about it than any of us do. Somehow I don't think they have been asleep at the switch.

Despite the anti-American frolicking of their college joes, I think the SK's will be up to the task. If the performance of the ROK Marines in Vietnam was any indication, the North can expect no mercy at their hands.