I think if Harold Godwinson managed to beat back the Normans and the Norse, we'd be looking at the most stable realm in the British Isles ever up to that point. Earl Godwin was the biggest landowner in England and Harold inherited most of this. He has the power to reward or pay off people with lots of land. His runner-up was Morcar, whose daughter Harold had married so they had an alliance.This - and it ties in with the "castle topic" you raise later - is another thing that needs consideration. Just how "stable" would this Anglo-Saxon monarchy be, how would its conflicts transform (for we can't just assume that in Late Middle Ages they take on the same forms we know from the Early Middle Ages of OTL)?
We may see a revolt of Edgar the Ætheling, we may not. Harold might just marry one of his daughters to the last known male member of House Wessex, maybe Edgar spawns some sort of cadet branch of House Wessex with very little power, maybe he ends up becoming a monk or a corpse, as frequently happened. I think Edgar the Ætheling may have some overlap with the English Investiture Crisis, so that could form a good reason for Edgar to revolt.
Harold had decent relations with the middle nobility of the north, as we see with how he handled Tostig. I think when Harold dies, though, there's going to be an absolute shitstorm because he had a lot of sons from two different mothers. This could range from a couple of short wars to a dynasty-destroying dogpile, with Morcar & sons supporting the sons of Edith of Mercia, some loyalists supporting a son of Edith the Fair, and maybe a wild card or two doing some wild card stuff.
The King of Denmark also had a claim on England. Denmark was something of a center of power, but compared to an England at peace it was a paper tiger. I think we could expect some alliances between the sons of Harold and Denmark, perhaps with the dynamic of North Sea power being turned on its head. If we really want to force a Hundred Years War analogue here, it's absolutely going to be England vs Denmark for who asserts hegemony over the North Sea, but that's pushing it. Regarding the Hansa, that's an interesting story because it partially rose out of a similar league in Visby going tits up because of a number of factors. One wonders how a lasting North Sea sphere would affect the growth of Baltic Trade Leagues. iirc, the Anglo-Saxons had some pretty unique trade methods that could have utterly transformed trade in the North Sea, though I need to delve a bit more into that. But yes, absolutely, we can expect medieval trade leagues to be a massive part of English trade in the 13th century on up.That would mean interference in succession disputes and the like? Or more?
How about the Northern parts of the HRE - they were economically closely tied to Scandinavia and England. IOTL, the commercial initiative lay there, e.g. with the Hanseatic League who sprouted contors from London to Novgorod. Any large trends foreseeable here with an Anglo-Saxon monarchy focusing on its role as North Sea hegemon?
I could see England aligning closely with the HRE over an investiture crisis, which may put it at odds with Denmark simply by association with the HRE, though to what degree I'm not sure.
Another thing to consider is how the Normans in France would fare. Would this open up opportunities for adventurer who otherwise would have gone to England to instead try their hands in Sicily, possibly solidifying a lasting Norman hegemony?I find both paths plausible and interesting to pursue...
I'll have to think of that. I'm pretty sure you're right here. That would require checking in which key battles English knights played crucial roles. Currently no time for that, but it could escalate butterflies into the Eastern Med much faster than I had thought.