We would need a POD in the the early 19th century and a different occupation of Denmark and WWII. So a POD and a century without buftterflies afterward.
So let’s say that the Icelandic national awakening hit 50 years later and are far less successful. So instead of two decades constitutional crisis about Icelandic relationship with Denmark, Iceland simply accept a full integration into the Danish state. We likely see Reykjavik turning into a Danish speaking city, after the loss of Schleswig-Holstein, Denmark invest heavily into develop Iceland much as it did in the Jutish moorland. As WWI hit and Denmark are isolated from its oversea possessions Reykjavik administrate them ad how. After the war Denmark plan for this better in case of a new Great War. As WWII break out in 1939, the Danish crown prince and his family travel to Iceland. As Denmark are occupied the Iceland set up a free Danish government and the crown prince is made regent of Free Denmark which includes Iceland, Faroes Island, Greenland and the Danish West Indies. The war ends and core Denmark end up on the wrong side of the Iron Curtain. Free Denmark renamed itself the Danish Commonwealth and keep claim to Denmark, but de facto are a greater Icelandic state. When the Iron Curtain falls, reunification are discussed, but the Danish Commonwealth are too small to afford a reunification with the much poorer Denmark and Denmark aren’t interested in becoming a monarchy again, preferring to stay a republic. As such the two states divorce official in 1992 on good terms.