Even though it was a very difficult battle against an already worn down enemy, I still think that William's army would eventually defeat Harold, though it could be a more protracted campaign to do so. Harold's strategy with two armies bearing down on him has to involve bringing each army to battle immediately and decisively defeat them.
William wouldn't immediately care about Harald and Tostig and if they're attacking the north. Since William would have to campaign there anyway, if two potential rivals can wear each other down, it works to his advantage.
The problem William's got is he's cut off from potential resupply and is operating in hostile territory so while he can do damage actually it's William who's on burrowed time and needs the quick win. So the longer any of it goes on for the less likely it is for William to win.
William also has the problem that a chunk of his army is made up of people only there on a promise of loot, land and titles, the further away that promise looks to being realisable the more likely William's key troops will find 'something better to do'.
In OTL I can see why Harold went to to meet him, but I can't help but think he'd have been better served consolidating his forces after Stanford Bridge and then harrying William until finally defeating him. Every day that comes will be more troops for Harold, and a day more William is trying to maintain a cohesive force.
Anyway William landing first will make life easier for Hardrada if Harold takes time to go after William first (which it's hard not to do otherwise he's leaving southern England undefended). IMO Hardrada can consolidate and operate better in the north than William can in the south, due to better logistics and more likely local support.
OTL is a great time line for Harold and he almost pull's it all off, he just get's unlucky at the end!