At the beginning of World War II, Iceland was a sovereign kingdom in personal union with Denmark, with King Christian Xas head of state. Iceland officially remained neutral throughout World War II. However, the British invaded Iceland on 10 May 1940. On 7 July 1941, the defence of Iceland was transferred from Britain to the United States, which was still a neutral country until five months later. On 17 June 1944, Iceland dissolved its union with Denmark and the Danish monarchy and declared itself a republic, which it remains to this day.
German interest in Iceland in the 1930s grew from nothing at all to proportions found by the British government to be alarming. The Third Reich’s overtures began with a friendly competition between German and Icelandic football teams. When the war began, Denmark and Iceland declared neutrality and limited visits to the island by military vessels and aircraft of the belligerents.[1]
Following the invasion of Denmark on 9 April 1940, Iceland opened a legation in New York City. Iceland, however, unlike Norway, did not closely enforce limitations within its territorial waters and even slashed funding for the Icelandic Coast Guard. Many Axis merchant vessels seeking shelter within the neutral waters around Iceland were sunk by Allied warships. The Chief of the Capitol Police Forces, Agnar Kofoed-Hansen, started to train the National Defence forces in early 1940.
The British imposed strict export controls on Icelandic goods, preventing profitable shipments to Germany, as part of its naval blockade. London offered assistance to Iceland, seeking cooperation “as a belligerent and an ally”, but Reykjavik declined and reaffirmed its neutrality. The German diplomatic presence in Iceland, along with the island’s strategic importance, alarmed the British.[2] After a few failed attempts at persuading the Icelandic government by diplomatic means to join the Allies and becoming a co-belligerent in the war against the Axis forces, the British invaded Iceland on 10 May 1940. The initial force of 746 British Royal Marines commanded by Colonel Robert Sturges was replaced on 17 May by two regular army brigades. In June the first elements of “Z” Force arrived from Canada to relieve 
the British, who immediately returned to the defence of the UK. Three Canadian battalions — the Royal Regiment of Canada, the Cameron Highlanders and the Fusiliers Mont-Royal — garrisoned the island until drawn down for the defence of the UK in the spring of 1941, and replaced by British garrison forces.
On 7 July 1941, the defence of Iceland was transferred from Britain to the (still officially neutral) United States, by agreement with Iceland, and US Marines 1st Provisional Marine Brigade replaced the British. Iceland’s strategic position along the North Atlantic sea-lanes, perfect for air and naval bases, could bring new importance to the island. The 1st Marine Brigade, consisting of approximately 4,100 troops, garrisoned Iceland until early 1942 when they were replaced by U.S. Army troops so that they could join their fellow Marines fighting in the Pacific.
Iceland cooperated with the British, Canadians and the Americans but officially remained neutral throughout World War II.
Reference> https://en.wikipedia.org/wiki/Iceland_in_World_War_II