As alphaboi says, the King and his family remained in Denmark - with the Danish Government maintaining an uneasy truce with the occupiers, at least for some years, the head of state could hardly go missing. Besides, one imagines that the historical precedent was weighing pretty heavily on him: In 1657, when the Swedes were about to lay siege to Copenhagen, King Frederik II uttered his famous words: “I will die in my nest” when it was suggested he evacuate.
The King was thought to embody the “cold-but-correct” stance towards the occupiers - a famous example being his response to a very long 1942 birthday telegram from Hitler. The King responded simply: “Meinen besten Dank. Chr. Rex” - a calculated snub that noone could call less than correct, but which none the less triggered a diplomatic crisis.