Interestingly, after Harold’s death, and during the mop-up phase, William was nearly killed by a small group of Saxons. In OTL, these Saxons failed to kill William before they were killed. A small tweak could shift this to having them successful before dying.

At this point, Eustace of Boulogne was badly wounded, and wouldn’t be available to command. Odo had been conspicuous by his presence well to the rear of the army, and his reputation had suffered as a result. There are a number of potential leaders in the immediate aftermath of the battle, but none with a clear case. Some, such as Robert de Beaumont, had proven themselves to be strong leaders, but without much in the way of men and/or money. Some, such as William of Evreux, had contributed significantly to the force, but were too young. And there was Robert Curthose, who was 15, and probably would be considered a little young to take control at such a critical juncture.
And, to remind, absent back in Normandy, so even less available than Odo.
The only thing I can see happening at this stage is the potential leaders having a council to decide on who is to be in charge.

Meanwhile, on the Saxon side, the Witan is going to be desperate to find a strong leader who can put up a defence against the Normans. Edgar had been passed over for Harold because Edgar was considered too young and not a proven warrior. Edwin and Morcar are still around, and their sister was Harold’s widow, so they’re probably next in line. Historically, they trundled south to oppose William, and that’s more likely to happen here. Their forces are still recovering from the damage taken at Fulford Gate, but they’re not likely to be disorganised.

On the other hand, the Norman claim to the throne has been rather damaged, and the leadership is unclear. Some of the Norman forces are going to be thinking about the need to go home and protect their lands in Normandy; Philip of France had been keen to put more control over Normandy, and with William gone, that’s a factor that’s going to increase. That said, many of the Normans had gone on the English adventure to gain English land, and younger sons without land in Normandy will want to stay put.

How this struggle plays out is anyone’s guess. One can argue that the disunited and disintegrating Norman army gets finally thrown back, and Edwin and Morcar get to decide who takes the throne of England. One can argue that the still strong and battle-experienced Norman army overcomes Edwin and Morcar’s fresh but relatively inexperienced forces.
So, the obvious alternatives to trying to conquer the whole England:
  1. negotiate for a Danegeld. But will Edgar/Edwin/Morcar agree? They might promise Danegeld, and then delay payment for their own bad faith, or because the English are not keen on obeying them and paying up. And even if paid, who gets the Danegeld? Robert and Odo to the rear, or the younger sons who came to get land?
  2. conquer just a part of England, like Sussex. How would they define the border between Norman England and English England?
 
So, the obvious alternatives to trying to conquer the whole England:
  1. negotiate for a Danegeld. But will Edgar/Edwin/Morcar agree? They might promise Danegeld, and then delay payment for their own bad faith, or because the English are not keen on obeying them and paying up. And even if paid, who gets the Danegeld? Robert and Odo to the rear, or the younger sons who came to get land?
  2. conquer just a part of England, like Sussex. How would they define the border between Norman England and English England?


The obvious breakdown would be along the old Danelaw. Whether that's a viable breakdown is a separate issue; Edwin and Morcar are already in charge of Northumbria and Mercia, which basically forms the Danish part of the Danelaw.
 
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