Suppose Harold keeps the high-ground advantage he has over William, and defeats William in battle. What happens next? Is William driven all the way back to the coast?
He's probably killed in battle, as nearly happened IOTL anyway: That's one of the likeliest causes for a Norman defeat.
Perhaps even sees an invasion of his own territory? Will Normandy be conquered and returned to the English, or will it become its own state? Or perhaps the French will take it?
France might try, but William's brother-in-law the Count of Flanders would probably intervene to help his relatives and the King of France wasn't really
that much more powerful than any of the dukes or the more important counts (such as Flanders) so William's oldest son Robert -- who'd actually have been his
only son if he did indeed fall in action at Hastings -- would probably end up as duke with about as much independence as William had possessed.
What about England? How will England's leadership be affected, and what might Harold's lineage do differently? Will a document like the Magna Carta ever originate in England? Will it ever be written at all (in any location)?
More contact kept with Scandinavia, less French influence and less involvement in French affairs (because, for one thing, there's a fair chance that this butterflies away the Hundred Years War...).
While Saxon England would not be as France-focused as the Normans necessarily were, France will still be important as it was nearer, richer, and potentially stronger than Scandinavia. By the time of the Norman conquest, Scandinavian influence in England was waning - Edward the Confessor was criticised for the Norman influence in his court. Saxon England will have the benefit of choosing which political sphere to involve themselves with at which point - they might also take on more Dutch/German influence than in OTL.
As you say, Edward was
criticised for that... and, as you
didn't say, the facts that Edward was not only half-Norman himself but had spent a couple of decades as an exile on the continent too were major factors in his bringing that influence over here in the first place and were factors that (in reaction to the invasion attempt) probably wouldn't apply for at least the next two or three generations of English kings.