This thread, while with a different PoD, may be relevant.
Basically :
William would have less chances to win there. Quicker preparation would probably have meant less men on his side, when Harold would have beneficied from more fresh and important ones, giving that Harold holdings were in southern England : once his army is gathered, it would be simpler to take on the Normans, not suffering the losses and fatigue as coming from Stamford Bridge.
Once William out, I think that (as Harold is still in his core and well held territories), the losses could have been relativly easily replaced, making Haralda's victory not that obvious. Indeed, northern territories weren't really controlled by Godwinson.
Victory wouldn't be as easy than IOTL, though, in spite of Harold's military and commanding skills. A good part of the Saxon victory was based on surprise and preparation after all, and Haralda would have gained time preparing himself.