First off, there is no reason to assume that King Harald Hardrada would not have rested his men for two-three days, and then defend his realm from the invasion of the Duke of Normandy.
Well, one has to take into consideration that it was not his home turf. He'd come in ships and would there be enough supplies to take on a march? Or would they sail down the coast? Split the forces to have some guard the ships or sail them? Would messengers with the news of William's landing have gotten a chance to tell Hardrada the news? The absence of range weapons/bows on the Norwegian side would still lead to a disadvantage. There are many factors to take into account.
The battle may not have been in Hastings, but would have been in the South of Endgland, excepting that his forces would have come off a fresh victory against the English Harald, and they would have been well rested now.
Stanford Bridge is about 250 miles from Hastings, for information's sake. How well rested would they be marching that far? They wouldn't have had enough horses for everyone to ride. Even rested, having to travel to find William's forces would take time and that would give William opportunity to find advantageous battle land or to set his troops.
Have you read much on the Anglo-Saxons and the Norse?
Secondly, of course there had been plenty of interaction between the countries you mentioned. However, after Hastings the interaction slowed and then stopped with the exception of interactions with what is now France, as the Normans did what many conquering powers would do: consolidate their power by limiting contact with foreign rulers. So, England was forced to gaze towards France, while under the Danes England would have been consolidated and gazing towards Denmark and Sweden.
Maybe, maybe not. I think that it is misleading to say that "what is now France". At that time it was a number of different areas and was not consolidated. Burgundy, the Acquitaine, Normandy were not united, though what became the Plantagenet line in England acquired vast holding through marriages such as that of Henry II to Eleanor of Acquitaine.
Another factor that applied to focussing to the south was what is called "The Little Ice Age" that made for more ice in the northern seas and shorter growing seasons. Things that once grew locally now had to come from the south. Did you know that wine grapes once grew in England?
Imagine the effect on the language we speak alone, as well as potentially not having the 100 years war.
I can give you a couple of ideas on that here. The author, Poul Anderson, once wrote a short essay on atoms using only words that would have come from the Anglo-Saxon/Norse/Germanic line "Uncleftish Beholding". Here's an excerpt:
"The firststuffs have their being as motes called *unclefts*. These are mightly small; one seedweight of waterstuff holds a tale of them like unto two followed by twenty-two naughts. Most unclefts link together to make what are called *bulkbits*. Thus, the waterstuff bulkbit bestands of two waterstuff unclefts, the sourstuff bulkbit of two sourstuff unclefts, and so on. (Some kinds, such as sunstuff, keep alone; others, such as iron, cling together in ices when in the fast standing; and there are yet more yokeways.) "
It can be found in its entirety here: http://www.grijalvo.com/Citas/Peculiar_English.htm
Then there's what computer terms likely would have been in Anglo- Saxon here:http://www.u.arizona.edu/~ctb/wordhord.html
Would Agincourt instead have occurred in northern Germany against a different foe? What would the Viking rulers of Scandinavia have done if they could have used English longbowmen in their aggressive armies?
Would the English longbow even have developed under a Norwegian/Danish rule? It evolved in the 1200's and onward in response to dealing with the Welsh. It was not an Anglo-Saxon weapon. http://www.britannica.com/eb/article-57608
There are many unknowns and 'might-have-beens' here. It is interesting and fun to discuss and speculate. But we can't know for sure what *would* have been.