As in OTL, 1261/62/64 Iceland became Norwegian together with Greenland. Now their fates were bound together: Since 1365, it was in personal union with Scotland; and when Norway became a nobles' republic in 1506 and again in 1562, Iceland followed suit.
In 1783/84, the Eruption of Laki volcano happened in Iceland. Half of the population died or emigrated, to Atlantis or Australia (OTL South Africa). This seems like a footnote of history, but descendants of said Icelanders often claim that they helped spreading the idea of radical democracy in German Atlantis, Canada and Australia.
In the Scandinavian elections of 1879, a coalition of Social Democrats, Socialists and left republicans gained the majority. Germany feared the Socialist threat, invaded, toppled the king and made Scandinavia a satellite republic.
The British Socialists also interfered, occupied Iceland and the nearby Färöer islands.