"Disobeying orders" is a rather extreme way of characterizing the political differences of Allied high command in 1945. Although nominally a subordinate of Eisenhower, Montgomery was a British general with more access to the British political system than an American general, like Bradley for example, would have to the American political system. If a decision Eisenhower made was adverse to British war aims or prestige (and Montgomery often conflated his personal prestige with that of the British army and nation) then Montgomery had a right to have the question settled by the political echelon above Eisenhower. This was always a sore point for the American generals, Bradley and Patton in particular, but it was a relatively effective system that kept the Alliance functioning with only a little bickering here and there. Perhaps the politics interfered with the Military decision-making in allocating supplies to the British led Market-Garden offensive over the American led Lorraine campaign, but it wasn't terribly disruptive to the war effort overall.

Anyways, on to Denmark. Denmark was "allocated" to be occupied by the allied forces in 1945 by the Yalta conference. In May of 45 it wasn't known for sure that the Soviets wouldn't try to occupy Denmark in a fait accompli, and influence it's postwar government. The British generally had a more suspicious view of Soviet intentions. Denmark controls access to the Baltic, an important Naval region for the Soviet Union. To Britain, this is much more of a security concern than the U.S. since British security traditionally relies on Naval dominance of the North sea and English Channel. To the U.S. Denmark is not notably more important strategically than other smaller European countries.

So what did the British armed forces actually do? Well basically they rushed into Denmark after it became clear that German resistance was dissolving. They used risky airborne, seaborne and commando operations to establish a nominal presence that would deter the Red Army from attempting to occupy the country, and pushed ground forces into Wismar (which had been allocated to the Soviets at Yalta) as a way to signal to the Soviets that Denmark would be occupied by British forces.

Eisenhower was tasked with controlling the meeting of Allied and Soviet forces in 1945, and generally took pains to prevent any friction that could be avoided with a more conservative approach. The British dash to Denmark was more risk than he would have preferred, but Eisenhower would never have risked a public spat that would have ensued if he had made an issue of it by accusing the British of disobedience after Denmark was occupied.