"How old is he when Edward dies?" is the first question.
If he's old enough to take the throne (not hard, Edward died at 63, plenty of time to sire a son old enough to take the throne in his own right even if we set the POD as of his marriage twenty-one years earlier), 1066 is the First Year In the Reign of Edmund* the Unremarkable. You might see Harald Hadrada invade anyway, William of Normandy doesn't have even the figment of a claim, however.
If he's not, he's probably put aside like his cousin Edgar (Edward's closest blood kin), and probably fares similarly.
* Its a pretty well established royal name with some dignity, so I'm using it for the hypothetical kid.