I was in Iceland in 1995, NATO training exercise when I was in the Army. Plenty of water, but like Dusty said, not much is public.
While out roaming around the countryside, I did see a decent amount of folks fly fishing, and when I talked to the “guides” they explained to me that most of the water was private, pay for system specific to each watershed.
Like any inquiring fly rodder, I did ask about “bug life” and patterns, and it was pretty general that most were fishing small stuff, Midges! Most of the Browns were Sea run fish, and they were typically keyed in on the abundant midges.
I was there in July, for a month. Absolutely beautiful country, about 20 hours of daylight, mornings were calm, with “large” black flies, swarms like in northern Canada, and they ate you! About noon time, winds would pick up and the black flies got blown away. Winds were quite strong all afternoon till about midnight. Temps were from 25-30 overnight, daytime went up to about 55.