Question: Has Britain's 1940 invasion of Iceland been downplayed by historians?
Yes down played by history, Not comparable to Germany's land grabs, Played a large role in WWII.
After Denmark fell to the Nazi's(April, 1940) Britain became concerned that Iceland a then Danish territory if occupied by Germany would be a strategic threat to the North Atlantic convoys which helped supply Britain important war materials throughout the war. When overtures for a friendly occupation were rebuffed, Britain invaded(May 10, 1940) in what turned out to be a bloodless seizure. Their takeover of Reykjavík, iceland's capital and largest city, consisted of posting a single guard at the post office and posting a notice that Iceland was now under British occupation on the door. The rest of the 746 British invasion force went to secure other parts of the island (Iceland's telecommunication service, the broadcasting service, the Meteorological Office and the German Consulate). Iceland protested their neutrality was being infringed, the British promised to leave Iceland after the war, and promised full restitution of all property damaged during the invasion and subsequent occupation. Then a week later, the British withdrew(May 17, 1940) and turned the administration of island over to Canada. Ultimately the United States took over administration in May 1941 about a year after the initial invasion before Pearl Harbor and the U.S.'s entry into the war. After the war Iceland occupation troops were withdrawn and Iceland did become an independent republic, a NATO Member, and "heavily integrated into the European Union". It's primary military base Keflavík (near Reykjavik) has been an American/NATO Navy base since 1951 when Iceland joined NATO. It was closed in 2006, but had been reopened by the United States when I visited Iceland in july 2019.
Germany attacked peaceful neighbors (Austria,Czechoslovakia and Poland) who were not at war at the time of the Nazi's invasions. Iceland was a possession of Denmark and Denmark had fallen to the Nazi's; and a new Pro Nazi government had been established in Copenhagen. Iceland was still technically answerable to that government and thus a legitimate war target for the British.
The British invasion of Iceland was never a land grab. The British from the very beginning stated their occupation was temporary. Not comparable to Germany's violent land grabs. Germany annexed Austria(March 12, 1938). There was never going to be an independent self governing Austria ever again if the Nazi's had their way. Germany annexed Czechoslovakia (September 30, 1938). Germany annexed the Sudetenland. Same for Poland in the fall of 1939. Germany immediately annexed West Prussia, Poznan, Upper Silesia, and the former Free City of Danzig. The UK never even governed Iceland, they declared from the beginning they weren't there to interfere with the internal workings of Iceland and left the domestic government in place.
The UK's plan was always about securing their important trade with North America and to keep Germany from establishing a base in Iceland which would threaten that supply route. The fact that the UK immediately transferred administration of Iceland to another country (Canada) and ultimately to the neutral United States in the first year, speaks to a world of difference between the Nazi's contested and bloody invasions / land grabs... and what the British did.
Played a large role in WWII.
I would say Britain really had little choice. A German Naval and airbase in Iceland would have been a catastrophe for the British war effort. The North Atlantic trade passage was the primary way war materials reached Europe from North America and that material was vital to not only the British war effort but also eventually the Soviet Union's war effort. It could easily be argued that a sustained disruptive Nazi naval and airbase in Iceland changes the outcome of WWII in Europe.