So I ran some numbers and this is basically my scenario:
If Denmark and Norway united into a binational state and consolidated all associated territories into a unitary empire state while building out their military and acquiring nuclear weapons over the next several years, they could potentially be a military power around the strength of Thailand - higher than Sweden and Algeria but below Saudi Arabia, Vietnam, Taiwan and Poland. So about 25th on GFP. They could also potentially catapult their submarine and frigate fleets to around 5th and 10th largest in the world, even though their fleet itself wouldn't be large - around 100 vessels, or about the size of Iran, Honduras and Malaysia. Their air power would also be small, they would have a fighter aircraft fleet comparable in size to Finland or Uzbekistan, but they would have a surprisingly large dedicated attack fleet for a European military, around the size of Belarus or Poland. Their total manpower currently sits around 150,000, but that could easily approach double that within a few years. They could also pretty easily acquire armor and attack helicopters from Russia. All-in-all, I see them comparable to a weaker European version of Israel in terms of their military. Culture and religion would of course be about the same.
So, basically once Denmark acquires nukes in 2026 and builds their military for another year or so - a Danish Missile Crisis erupts in summer 2027 when the Danish Crown orders nuclear missiles to be stationed in Greenland which results in Denmark and NORAD standoff and posturing, but again - I feel like there are too many deconfliction mechanisms for a full-blown war between Denmark and the U.S./NATO. More than likely there are some Arctic skirmishes in Greenland or Svalbard that result in the deaths of a few dozen civilians and few hundred combat deaths - WORST case scenario. I think a low-level guerilla war and infiltration into neighboring Holland, Germany or Iceland to hold expansionist territory. Sweden is too militarily frail to resist and is culturally close to this new state automatically, so Sweden or Finland going to war with this new empire is also out. Germany would most likely treat this new threat similar to organized crime and retaliate with legal and local law enforcement actions. Poland and Denmark-Norway are probably too far apart to real do much fighting. I see this conflict largely contained, isolated and defused within a few years or decades.