I am currently taking an undergrad class on Modern Iran and was rather surprised to find that Iran was invaded by Britain and the USSR during World War II. The invasion wasn't covered in detail in my class readings; I'm curious as to how the invasion worked on a political and military (on-the-ground) level, as one of my main takeaways from the reign of Reza Shah was that his army was a large, well-funded, relatively "modern" one. I don't quite understand what factors definitively contributed to its defeat by the Allied forces.
Hope you don't mind a delayed response.
Britain and the Soviet Union both claimed to fear the presence of large numbers of Nazi fifth-columnists present within Iran. The country already had plenty of reason to hate Britain and Russia, being regularly used as a plaything by both in the 19th century, and the large numbers of Reich citizens, working in key areas such as communications, were thought to be part of Berlin's master plan to subvert the country; that they would find fertile ground in Iran was not doubted even if Iran was nominally neutral. The Allies invaded on 25 August 1941. However, even before the war ended people were casting doubt on this story--basically saying "come on: secret Nazi hordes? Surely you could have come up with something better". I've reviewed all the British primary source materials on this and can assure you that that's what British planners believed, even if it turned out to be a joke (there were far fewer Germans present than suspected, and they were nowhere near as well organized as suspected).
The invasion of Iran was conducted in two very different fashions. In the north, the Soviets came in with overwhelming might. In the south and west, the British force relied more on mobility and aggressiveness (and the simultaneous Soviet invasion) to counteract their opponent's overall superior numbers. They did not always have better equipment than the Iranians (who did put up a fight), but the British Gurkha and Indian forces were making progress (and, most notably, had secured both air superiority and the AIOC refineries at Abadan) when the Shah, who was convinced that the Allies would not invade, threw in the towel three days in, though some fighting sputtered on past this point. The collapse of Iran was thus due primarily to morale factors, but I think it's pretty clear that the Shah knew he had no hope of standing against what was being thrown against him. His army was fighting, but was being defeated, and there were signs of a collapse of military morale.
If you want an overview of the invasion, I'd suggest Richard A. Stewart's Sunrise at Abadan. It's pretty much the only modern work on the subject as a whole, though there's a series of specialist writings, mostly involving the intelligence side of things, by Adrian O'Sullivan.