Alright, I got it:
Iceland does not secede from Denmark in 1944, and instead remains, albeit grudgingly, a part of the Danish kingdom. However, postwar, after promises of independence from King Christian X come to naught, they rebel in 1945. A short, yet bloody, conflict in Iceland from 1945 to 1947 causes Icelandic independence under a republican constitution. However, communists overthrow the interim republic in a coup, with the popular support of Icelandic nationalists. Thus, the People's Republic of Iceland is formed in early 1949. The Icelandic Army is formed as a compulsory service force for the defense of the Republic. However, in an attempt to curry favor with the other powers, it slides away from Russian influence, and join the Non-Aligned Movement.
The PRI persists until 1976, when the conflict with Britain over fishing rights and territorial waters erupts into full-blown war between Iceland and the United Kingdom. After a bloody three-month conflict, Iceland is occupied by British forces, and eventually restored as a parliamentary democratic republic in 1981.