Also how much of the BEF are you looking to have go on this operation? The forces that OTL deployed to France?
Probably most of it. IIRC the BEF was 4 divisions plus a little extra in August 1914. When Germany occupied Denmark in 1940, they devoted 2 divisions, but they were doing it under more favorable circumstances, so doubling the forces involved seems like a starting point if the British want to do it.
2) landing in friendly ports not hostile ones.
Point well taken, but I’m curious how much resistance the Danes would offer. If a Royal Navy squadron shows up at dawn and the ambassador hands them a note telling them to surrender Copenhagen’s fortifications or the city gets bombarded, what do the Danes do? The 1940 Danes folded, but the 1914 Belgians fought. However, even if the 1914 Danes do fight, do they even have the ability to hold off an amphibious landing until the Hochseeflotte arrives? Worst case for Britain is a failure like the attempts to force the Gallipoli narrows: mission failure with multiple large ships sunk due to mines and coastal artillery. But best case is like what happened for the Germans in 1914: Danes surrender by noon, and their ports become bases for further operations.
The German's will have the ability to crush the British beach head on the Jutland with relative ease..
I think you’re actually right so I don’t think the British would devote most their attention to Jutland. What the British care about the most is command of the Danish straights, which means Zealand (with the large port of Copenhagen and potential command of at least the Oresund) is the real priority. Funen is probably a priority too because of its position on the Great Belt. These are conveniently islands with a defensive advantage. If objectives in Jutland are included at all, (IMO) the safest options are landings in the North Jutlandic island to neutralize the port of Aalborg and establish a defensive line along the Limfjord.
You'd also be handing the Danish navy
I don’t think the Danes had a navy that was worth much. Some obsolete coastal defense ships and a handful of torpedo boats maybe?
The effect on the USA of Britain carrying out an invasion of a neutral nation i.e. doing exactly what the Germans did in Belgium, will be huge. Propaganda score draw.
I think propaganda is worth something, but in 1914 American intervention is a long ways away.
The latter will benefit from having their short-range small TB's available on their own doorstep
If (and this is the if that the operation hinges on) the British take Copenhagen as the opening move of the campaign, the Royal Navy will also have a location for their light forces to coal in-theatre.
It is not just an issue of the RN having the ships to cover, it is having the ships active. Ships need to go into maintenance and refit, and with the GSF on constant activity (and unsure how well Copenhagen can support them) if they ever pull the GSF back the HSF is going to sweep any light elements left behind which basically forces the GSF to stay on station away from their facilities.
I think you’re right that the Grand Fleet proper would not operate out of Copenhagen. But substantial light forces could. As long as there’s some fortifications to hide behind, those Royal Navy forces aren’t going to get swept away any more than Goeben did(n’t) at Gallipoli.