Let's assume the man fell in battle, and King Harold II is still alive. What could the economic and political realities look like in Britain and Ireland in the ensuing decades and centuries as a result? I know centuries is really far out, but I'm looking for different scenarios that could very plausibly happen, not what one thing is the most likely scenario.
Well, the general assumption is that England would have been more focused on North Sea than Channel and Atlantic, at least for a moment.
Saxon influence, and more broadly HRE's influence, could be as much present than the Frankish one before the conquest, with all that it implies culturally.
Economically and Politically, late Anglo-Saxon England knew a similar process than what happened in the continent : desintegration of the kingdom into smaller independent political entities (unified by a common kingship). Earldomancies would be a probable base for these to appear, on the ground of old AS entities (Bernicia, East-Anglia) or late AS subdivisions (Western Mercia, etc.).
It doesn't mean this
AS feudalism would be similar to Norman one, of course. As mentioned before, while Frankish institutional influence would certainly exist, Saxon (as in continental Saxon) would be another factor printing on a large local base.
It would be as well influenced, in its prime form, by a lasting slavery (that virtually disappeared elsewhere and probably will there as well) and by proper AS social identities as cotarii or bordarii.
A more important nobility, demographically speaking (closer to continental standards, between 4 to 6% instead to less than 1%), more diverse socially would count as well.
I think the traits of English institutions would have probably lasted, making *English feudality looking more a mix between German feudalism (huge, distinctive entities, with an important royal political role) and Frankish (lack of public lands, or conquest over non-Christians allowing the landed redistribution)
But at this point, it's more an educated guess.
Northern England would be, obviously, less damaged ITTL (while previous damages were due to Scandinavians), and form a relativly stronger economical factor in North Sea.
I think that this *England would still go for Wales (Harold and previous rulers having prooved their interest on it) and earlier than IOTL; and maybe Scotland, critically without Davidian Revolution (while, admittedly, a simimar evolution could take place, but slower than IOTL without the radical change that Normans were).
It would be even more true with an England that have little to no matters in France at all : once Norman/Angevine hemeon butterflied away, not only Capetians may know an earlier rise (while probably a slower territorial growth), but the whole of Western European international policies are butterflied (it kinds of prevent describing situation going on for centuries).