I was in Sweden in late January. It was actually not any colder than it was here (which is New York). In fact, most days it was actually warmer. Still below freezing, but when I left NY temperatures were in the teens. In stockholm they were steadily in the lower 30's/upper 20s. I imagine Denmark is about as warm as the southern end of Sweden.
But it depends on where you are going. In Kiruna, it's arctic.
A puffer or heavy wool coat, boots, and a hat should be fine if you're south. If you're going further north, you'll probably need something closer to gear. Una/AlaskaGirl would probably know more about it than I do.
(Side note: If your sweetie can get me a job in Sweden, I'd be positively over the moon. I've ached to live there ever since going there myself).