It wasn’t a conquest. It was in lieu of a dowry.
They were the security for the King Christian of Denmark paying 60,000 crowns of the bride-money for his daughter, the Princess Margaret.
Remember: this was in effect putting the islands into pawn — though the debt has never been redeemed. But that is because of the resistance of the Scottish and then the British authorities.
In the Treaty of Breda, 1667, the status of the Scottish tenure was “unprescribed and unprescribable” — which amounts to an admission that the status has not been changed — indeed, under treaty law, cannot be changed.
Two years later, 1669, Charles II tried to rationalise the situation with his Act of Annexation, which made the islands his personal responsibility:
It is not only fit in order to his Majesty’s interests, but will be the great advantage of his Majesty’s subjects dwelling there, that without interposing any other Lord or superior betwixt his Majesty and them, they should have an immediate dependence upon his Majesty and his Officers.
By the 1707 Act of Union, the islands were transmogrified into counties of Scotland.
In 1906 Norway become independent of Denmark. An official missive from Shetland went to King Haakon VII:
Today no ‘foreign’ flag is more familiar or more welcome in our voes and havens than that of Norway, and Shetlanders continue to look upon Norway as their mother-land, and recall with pride and affection the time when their forefathers were under the rule of the Kings of Norway.
Even into the late twentieth century the Scottish judiciary was wrestling to reconcile the status of the islands. Only in 2005 was the white-cross-on-blue flag of Shetland (the same banner as the Hvítbláinn of the Icelandic republicans) authorised by the Lyon King of Arms.
Even if the SNP were to win their referendum, they may find they have secession problems of their own.
For true wranglers, the issue is debated at length and in detail here.