Did not see the above post, so thought I would answer some of the questions to the best of my ability. Disclaimer this is on the top of my head, so details might be
Could the interests of the two kingdoms be kept separate, Britain looking west and south and Denmark looking east and south?
Denmark will likely keep looking south (I get to why later), east much less so. More importantly Denmark will eventually look west, no doubt about it. Furthermore Norway is already looking west, and Norway while at this point a mere "province" of Denmark and not a separate Kingdom. Is not without value (again I get to that in another answer)
Would the ATL king have to spend more time in Denmark than the OTL king spent in Hanover?
To be fair I don't know how much time the Kings spent in Hanover. But I do think the King of the Anglo-Danish union could get away with spending relatively little time in Denmark. A good suggestion for a royal governor or whatever you wanna call it for Denmark could be the bastards of the Danish kings. As a old and now banned Dane on this forum called the Gyldenløves (Surname given to a series of illegitimate sons of Danish Kings): They were embarrassingly loyal to the Danish throne.
Arguably the most famous of these illegitimate sons is still alive in 1714: General-Admiral Ulrik Christian Gyldenløve, Count of Samsø. Though he would die just a few years later in 1719, 36 years old.
Now Frederik IV of Denmark the King that would become King of the union also had a bastard son, but the young one did not survive to adulthood (born in 1704) It is certainly not out of the question that in this version of history, more bastards would be born, and as such you have access to a loyal cadre of well educated governors.
Would/could the Danes use British support to take back Scania? Would the Baltic powers ever rest if the Danes got back control of the Oresund?
The union would happen just two years after a Danish (re-)entry in the Great Northern War, and I don't see the British stopping Denmark from doing anything in the war, on the contrary.
The goals for Denmark during this war is sorted down below, with the first being the most important (IMO)
1: The Holstein question solved, Holstein-Gottorp is too much of a thorn in the side of Denmark, this was the main goal of Denmark in the war (it was fulfilled)
2-3: Bremen-Verden, during the war it was overrun by Danish soldiers in 1712, in return of George I's aid, Denmark sold the occupied province to the Electorate of Hannover. In this timeline that won't happen, Hannover is nothing to Denmark without the personal union, it will stay occupied by Danish soldiers and likely is ceded to Denmark in the peace treaty, bonus points for Denmark now that Holstein is in Danish hands, in control large parts of the coast, including the Elbe and Weser estuary's. Remember Oldenburg during this timeframe is Danish.
2-3: Skåne, the Dane in me wants to say that Skåne would be the second most important objective for Denmark, but I think Bremen-Verden is a lower hanging fruit, nonetheless I could still see Denmark pushing for Skåne. Sure The Netherlands will object, but heavily declined I argue that Denmark is the stronger of the two.
4: Swedish pomerania, Rügen and Wismar.
5: Gotland - Estonian isles etc.
6: Norwegian lands lost in 1658.
It's important to note that Denmark during this time is HEAVILY militarized having a large number of the population in the military, just as Prussia and Sweden had.
Would things be better if Sweden controlled Norway, perhaps as a trade for Scania?
No (IMO).
Three reasons.
1: Norway will eventually look west, if it does not already do it. This mean the Norwegian population will benifit more from staying with Denmark.
2: While often deemed backwards by Danes, Norway was the cashcow of the Dano-Norwegian Kingdom, not Denmark. (not counting the Sound toll) Loosing Norway while winning the great Northern war makes no sense. Furthermore it have valuable resources that Great Britain will drool over. Noticeable crucial naval supplies.
3: I think it was
@Jürgen that mentioned this a few months ago. The loss of Norway during the Napoleonic wars had a profound effect on Holstein and more importantly the emerging nationalism.
Before the loss of Norway, the center of "power" in Holstein was the eastern coast and trade in the Danish straits and Norway. While after the loss, the shifted west to the south-east, increasing ties with the emerging German states making sure German nationalism took roots in the 1840'tish. Eventually leading to to the loss of 40 % of the Danish state and 60 % of the economy.
Now as mentioned above I would go on to talk about which "way" Denmark was looking during this period. Traditionally it have been south to Germany due to the economic ties. Nationalism, the loss of Schleswig-Holstein and industrialization would change this. after the loss the UK became the most important trading partner of Denmark, but before the war it had already been a trend that trade was increasing with the UK, the war and nationalism simply cemented it. Due to this I argue that Denmark will at some point in the nearby future of 1714 change its trading patterns from the south to the west, keeping Norway and a union with the British isles will simply make certain it happens earlier than OTL (IMO)
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Anyways I hope it made sense

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