For all the noise Britain might have made regarding her moral interest in the brutality the Russians showed towards the Poles, they were not interested in doing anything more than potentially putting economic pressure on Russia. There was no reason to declare war and no coalition of willing allies to push the issue.
While the Shweig-Holstein crisis might have sparked a war if Prussia had been overtly belligerent (doubtful with a shrewd operator like Bismarck at the helm) you would have needed France (and most likely Sweden as well) to be willing to intervene, Britain would not go it alone, and considering Denmark was in the moral wrong in the eyes of many, it would have been unpopular. Which is probably worse from the eyes of Whitehall.