Well, if you start at the Viking/Middle Ages, the Scandinavian countries have had a concept of kings, nobility as far back as recorded history goes, but also of decision making in assemblies (Þingr). It was not outright elective monarchy but at the very least a king did need the support of the thing to rule, and the oldest laws had a 'constitution' of sorts in a "King's law" (Old Swedish kunungs balker) which delineated how kings were to be elected, deposed and their responsibilities. So it's important to note we didn't start as absolute monarchies up here. Scandinavia did not have a true feudal society in the middle ages.

In Sweden, the medieval thing evolved into the "Riksdag (parliament) of the estates". Formal absolute monarchy was really restricted to two periods: The first was the 'Carolinian autocracy' instated under Karl XI around 1680, which partially was a reaction to the fact tha the nobility's power had increased dramatically in the prior decades. This lasted until the death of Karl XII in 1718, after which the 'Age of Liberty' (Frihetstiden) began, and the constitutions of 1719 and 1720 put most power in the hands of parliament. That lasted until Gustav III's bloodless coup d'etat in 1772, which instated absolute monarchy until the constitution of 1809, from which point on the king's role decreased over time in practice (and legal precedent) to being a figurehead. This figurehead role was formalized in the constitution of 1974, but it had de facto been the case since long before (and in terms of legal precedent, de jure as well)

Both Swedish reversions away from absolute monarchy were responses by the aristocracy and bourgeoisie to military calamities - the death of Karl XII in the Great Northern War, and the 1809 loss of Finland to Russia. So they were aided by the fact that there already existed an aristocracy eager for power, and a parliamentary tradition.

Denmark had a single, longer period in parallel: Under Frederik III, the Danish Kongeloven of 1665 formalized royal autocracy and the Danish parliament would not reconvene for almost two centuries. The Danish constitution of 1849 was the end of it. Denmark's conversion was more in line with the rest of Europe, as constitutional monarchy was an idea who's time had simply come in the mid-19th century. As it were, other countries had already started the Revolutions of 1848 when the Danish king died, which created the opening necessary for Denmark's own reformers/revolutionaries to demand a constitution.

Norway was ruled by Denmark until 1814, when it entered a union under Sweden, under which it was ruled by its own parliament and constitution.