"Plausible Check: Effects on the British Isles from a Surviving Anglo-Saxon England, or both Harald Hardrada and William the Would-be Conqueror fail to take the English crown"
I am not sure as to how a surviving Anglo-Saxon England would have fared, had the two candidates in question failed in their quest to grab the English crown.
Tolkien would be very pleased, since he considered the Norman conquest to be the most horrifying tragedy in English history. (He wouldn't
exist, with a POD back in AD 1066, but you know what I mean...)
England would be considerably less centralised for the time being, with a less pronounced and 'removed' nobility. More lesser nobles who were still generally rooted in the local community, rather than high-up overlords who spoke a different language and considered themselves categorically different from the masses.
There would be no enforced centralised legal code, and no circuit judges. Legal matter would be ground-up affairs as well, following the traditional Germanic pattern of having a local judgement first, and then the possibility of appealing it higher up. (This was later restored, in part, both in England and later in the USA. But William did his utmost to crush this system: it gave too much liberty for the peasants!)
The tradition of representative bodies would also persist, also from the ground up. Decisions would mostly be made in local assemblies, and for greater matters, such assemblies would send the men they considered the best and wistest to form a greater assembly (the Witan, literally "the Wise"). To be fair, such assemblies would not be open to just anybody. Functionally only the land-holders, but we must again note that in Anglo-Saxon times, there were far more "little" land-holders, instead of a small clique of huge magnates. And the nobility was far more "open", instead of forever closed to the supposed "lower classes".
Finally, although not all kings would like it, the nobility would have a continued say in who gets the crown. Kings of England would be chosen, not born. More like the HRE. It would probably be men from one or a few major families at all ties... and often the son of the last King... but if an heir turned out to be unsuited, they'd be able to choose another.
All in all, you'd have a far more decentralised, localised and egalitarian country and culture. Politically less stratified, lrgally more rooted in the community. Infinitely more ground-up than top-down. Far more of the typical Germanic ways and institutions survive. The language remains Germanic, too: not some strange hybrid with loads of Romance influences.