And what if they join the war late (in 1917 after a more desastrous Nivelle offensive), and this put them in the winning side of WWI, which colonies could they grab?
Could they, perhaps, grab the Orkney and Shetland Islands? It'd require decisively defeating the Royal Navy, most likely, but having Denmark in the Central Powers puts the CPs in a much better position to do exactly that, just by having naval bases further north up the Bight and having complete control over both the Skaggerak and the Sound.
If they couldn't get those islands in the peace treaty, perhaps they could get Ösel/Saaremaa and maybe even northern/all of modern-day Estonia or Livonia? I don't know if Danish rule over Livonia stretched as far south as modern-day Latvia, but IIRC, modern-day Estonia, including Saaremaa and Hiuumaa, were under Danish control (albeit) in the medieval period.
Perhaps such an arrangement would be hard to administer, and it might seem implausible at first glance, but here's why Denmark receiving Livonia could work:
- ensures that the Danes are well-rewarded for their efforts
- ensures that the Danes have the opportunity to territorially expand in another direction, blunting any revanchism or ill will with Germany over Schleswig-Holstein
- ensures that they have continued goodwill towards Germany, ensuring their acquiescence in remaining in the Central Powers and joining a German-led customs/economic union
- ensures that they have a dog in the pit with regards to Soviet (Russian) attempts to regain Brest-Litovsk territories, etc., thus ensuring that the Danes will continue looking towards Germany for security solutions.
In my mind, it would be best for all sides involved (barring the Russians/Soviets, of course) if Denmark annexed both modern-day Estonia and Latvia. The population of those two countries is about 3.2 million OTL, and so would be big enough in comparison to the population of Denmark proper that the Estonians and Latvians could reasonably expect to secure minority rights/devolution/autonomy/federalism to keep them content, but not too big to where the tail starts wagging the dog.
Any separatist movements could be countered with the claim, "Leave us and you won't be covered by the Central Powers anymore, and so you'll fall prey to the Big Bad Soviet Commie Bear (TM)".
If Denmark couldn't receive Livonia, they could just have North Schleswig returned to them by the Germans. The Germans will probably making territorial gains elsewhere to make up for that, and the North Schleswig region didn't have any particularly savory natural resources, AFAIK.