Assuming Prussia actually intends to do this, I think the Danes will buckle down and really fight hard. Remember, Denmark was smaller than Prussia but its navy was still bigger, even then. Jutland would fall due to the enemy's overwhelming numerical advantage but Copenhagen is on an island and this could present a problem, though the Austro-Hungarian navy was on its way and would have been up to the job. Also recall that Sweden didn't come to Denmark's help IOTL. Prussian armies within sight of Scania might change their sympathies (or maybe just scare them). If they did come to their Scandinavian brother's aid, Prussia would have to fight an amphibious war against formerly friendly nations with superior navies. This time, Prussia would be in a bit more trouble.
To add to this, if the Prussian made their intention to annex Denmark clear you have to wonder if the Austro-Hungarians wouldn't become much less eager to pursue the war. Why would they continue to fight to enlarge Prussia (instead of just defending Germans), in a war that could very likely see an intervention?
No. None of the Prussian Royal family had a claim to the Danish Throne and I doubt that the Great powers would let Prussia annex Denmark. After all the Princess of Wales was a danish princess and the Tsarevich was engaged to another Danish Princess. So if Britain would not intervene than Russia would.
Yeah, intervention seems like a pretty realistic proposal. The Russians had already forced the Germans to give up in the first war*, and they could do the same here. I doubt either Britain nor Russia would like complete Prussian control of the access to the Baltic, and as Chris Triangle says, the Swedes might decide at that point that they have no choice but to intervene as well.
Russia and Sweden joining would severely tip the balance of the war, and I kind of doubt the intervention would even happen. The Austrians don't seem to have their heart in it to the same degree as the Prussians, so the threat of war with Russia might make them convince Berlin that it was time to end the war. I would expect Russia to be able to force a compromise without actually having to go to war.
I wonder what the effect of that would be, especially if the compromise was Holstein alone being annexed. Schleswig was after all not actually a part of the German Confederation. Maybe it could see some discontent directed at Prussia, since their desire to aggrandize themselves had screwed over fellow Germans?
Though really, all the reasons outlined above make it clear why the Prussians wouldn't (and didn't) go for it. Even if they somehow managed to conquer Denmark without intervention, they're still left with another bunch of people who hate their guts, who would likely be supported in their attempts to free themselves by other great powers. It also makes Prussia/Germany seem even more like a warmongering nation, which is probably not good for them in the long run.
*Which was completely ignored in Denmark, which made us think that
we had beaten them all on our own and could do it again, with tragic consequences.