I think a important aspect are that a Denmark regaining Scania, Blekinge, Halland and Bohuslen and gaining Gottorp and Bremen-Verden would be a great power, but it would pretty much be the same as Prussia anno 1750, the least among the Great Power of Europe.
So what would Danish goal be. Either expand east with a union with Sweden, which I think would the goal Denmark spen the least resources on, but Denmark would likely support a independent Finland in case of Russia trying to establish one.
Denmark turning inward and continue reforms, centralise the Danish state and enlarge army and navy, I think this will be main policy.
Denmark expanding its influence in north Germany (primarily through marriage), I think this will fill much more in Danish foreign policy than a union with Sweden.
Will this crash with UK, not really, a stronger Denmark are more likely to ally with UK, in OTL fear of France was the main reason Denmark didn't become a British ally. Here I think we will Denmark support the Austrians and British in the Austrian Succession War and join the war. The Swedes will likelly use the opportunity to declare war on Denmark and/or Russia, and I think the result will be Sweden losing Jämtland and Swedish Pomerania to Denmark, and a independent Finnish kingdom being set up. For Prussia the Danes siding with Austria will be a disaster. As they will likely be the main target of Denmark after Sweden have been defeated. I don't think Denmark will go after major gains. I think Denmark may take Strettin, but not much more than that.
Would it really be tough? Circa 1750 Prussia was able to field a 150K (the best of Europe by a pretty descent margin) and was, essentially, at the center of european politics. They where the defacto leader of most of Protestant Germany and where seen as the best means for the German princes to counterbalance any ambitions the Hapsburgs might have to make their imperial title means something.
In comparaison Denmark had a 40K army during the Northern War and a descent sized fleet. Quality-wise both where capable but nothing more. Sure, its likely the size of both will grow some ITTL but I doubt it would be enough to make the ensemble as military valuable then the Prussian army. Moreso, while it would indeed have a beachead inside Germany it would still be decisively on the fringes of things compared to Prussia
IMO it would be more a case of first among the mid-powers then last among the great powers.
As for what would happen there, I do believe you have good notions: a raproachement with Great-Britain would make allot of sense and any further expansion of Denmark influence would have to happen through diplomatic means, with Jamtland being a bit over the line in term of what could be worth it. More broadly, however, the butterflies would most likely be significant enough on the diplomatic scene of Europe that the War of Austrian Succession, if it even happen, would have a different enough setting that one would need to discuss it a bit furter before trying to imagine the results.
Generally speaking it is likely that ITTL Helsinborg would be far, far, far from the last battle in Scania. Sweden is severely weakened but isn't broken, it definitely can't pretend to be a great power anymore but it does retain some serious military capabilities in the mid to long term. The ITTL version of the Hats are probably gonna be far more focused on Denmark then Russia and I could easily see the Greater Scanian region and Bohuslan pretty much being to ITTL 18th century Sweden what Alsace-Lorraine was to late 19th-early 20th century France OTL.