for William, England was his inherited right- he wasnt going to concede anything.
In 1066, Northumbria's status as definatively 'England' is, in my mind, less than concrete.
However, I do agree with you about the fact that William will be under pressure to leave quickly in the wake of a defeat.
Then again, it depends upon the location of the defeat, the level of control the Normans have on the South, and what the Saxons are up to.
If both Gyrth and Leofwine (Harold's brothers killed at Hastings) die at Stamford Bridge, the Saxons may fall in line with Tostig behind Hardrada.
They may instead rally to Morcar, Edwin and Edgar Aetheling. What they would do is unknown. Tostig had been removed from Northumbria by Harold due to his unpopularity. That hatred may still remain.
If one or both of Gyrth and Leofwine survive, what they do will be interesting. Will they join Tostig, to take positions of power in Harald's new regome? Will they bid for the throne themselves as Harold's heir, or support another.
I think that what happens next is very dependent upon the fallen at the battle of Stamford Bridge and the subsequent actions of leading Saxon nobles.
There could be a three way war, the Saxons could side with either Norman or Viking, or divide between them.