What if Denmark possessed Orkney and Shetland during the Napoleonic wars, would Britain demand them from Denmark? Denmark OTL ceded the islands to Scotland as a wedding gift in 1469.
https://en.wikipedia.org/wiki/Shetland#Annexation_by_Scotland
They did not. The King of Denmark used them as collateral for a dowry, as well as getting cash out of some of the others. The treaties involving it had it where they could be bought back, but the Norwegians (the ones who owned the land the Danish king gave up) were refused by the Scottish king. Thing if someone breaks into your house (or a friend or relative does in), takes your stuff, and then pawns if off, you take the slip to the pawn shop and try to pay back what they owe, but the pawnbroker refuses to honor the agreement and keeps the stuff someguy stole from you.
Back to your questoin though, they might. At the very least they will want fishing rights, though politics in Britain might be against that. Might deflate prices. Actually, will the Danes have taken part in the Continental System differently? And I see a possibility of these islands being used by smugglers, who make a killing by getting around high taxes on luxuries and necessities. Do you guys think whichever island taken would be attached to Scotland or to be put in a situation like the Channel Island or the Isle of Man? Perhaps even a situation used overseas? I actually could see a good thread and conversation going for six or seven pages just on the subject of administration.
More than likely, however, I do think that parliament had no common sense. Instead of annexing Heligoland, why did they not at least annexe the Faroe islands? But the UK was becoming a world power so they could have taken Iceland, Greenland and Svaland. These islands were not much use then but in 21st century and in the forthcoming centuries they will. Politicians are short-sighted and have no strategic perspective, the same applies to the British invasion of the River Plate. (Patagonia)
Heligoland was right off the German coast, was a a conduit for bringing in German mercenaries for the British, and was fairly close to the Mainland possessions of the British royal family. Bismarck later slandered Caprivi by giving him responsibility for the Anglo-German treaty of Zanzibar, though it really was in Germany's favor as they got a large strip of land in a fertile area leading to major rivers, they got acceptance by the British of German rule over vast swathes of land the Germans hadn't really went into yet, plus they got an island that could have been used to start blowing up German ships immediately in WWI.
As for Svalbard, Greenland, Iceland, etc, those were and mostly remain frozen wastelands. Telling people to make money from volcanos, bringing tourists to the Arctic Circle, or digging beneath the ocean for some black gloop or hot air would have people think you were making a joke back then. Besides, it might have brought a liiiitle more hostility into Scandinavian and British relationships, given that the British were the ones who attacked Copenhagen, destroying their fleet, and basically forcing Denmark to join Napoleon unless they wanted to be conquered. Besides they, unlike then Prussians, didn't go about profiting as an ally of the French.