EDIT: actually the nuclear war is beside the point. Just that conventional shelling South Korea while having a number of deliverable nukes on hold might become a weekly routine for the North.
The first hour of a full-on war on the Korean peninsula would make 9/11 look like a day at the park and would generate more civilian casualties in an industrialized first world nation than any event since WWII.
The second order effects of losing South Korea don't bear thinking about, either. Everybody saw how fast the dominoes fell when 10% of one sliver of American mortgage holders decided to go late, right? If Seoul gets burned, several hundred billion dollars worth of high quality bonds held by primarily Japanese and American financial institutions burn with it. That would put the market into pure, unrestrained panic within literally seconds. It would get worse over the coming months as countries who traded heavily with South Korea (let's see: China, Japan, Taiwan, US, etc) saw their orders go to near zero and had their supply chains dependent on SK products start blocking on input.
Plus, in the middle of this, Obama gets to make a phone call to whomever takes over when the Japanese government falls and say "Listen, I know you're kind of spooked at the moment and have just been elected on a platform of We Must Stop North Korea, but please please please please don't announce nuclear capability despite that being within the capability of Japanese industry in a matter of weeks. China would get a trifle upset."
I would think the goal would be to save as many humans as possible (while, admittedly, probably not willing to trade South Korean lives for North Korean lives) and toppling the regime in North Korea as quickly as possible. I don’t know whether that includes nuclear weapons. I would hope that nobody would be willing to launch a nuclear weapon just to kill as many North Koreans as possible.
The deciding factor might be China's position on the matter. It provides quite a bit of support for North Korea and whether the USA would want to get dragged into a conflict that might escalate beyond the Korean peninsula remains to be seen.
Having said that I don't think there is a significant risk of serious conflict since it is in China's long term interest to keep some form of low-level conflict with the USA or its allies going in the area, if only to further its territorial ambitions with the Spratly Islands, http://en.wikipedia.org/wiki/Spratly_Islands and other parts of the South China Sea which apparently has significant old reserves.
And presupposes that North Korea choses the least self-destructive action.
Feeling lucky?