It's probable that Harald and William had some sort of agreement or at least half-hearted strategical alliance. The links between the courts of Normandy and Norway are known, even if there's not much to be known about, and the preparatives (as well the campaigns' schedule) could point that.
Now, what form would have taken this agreement?
I think it would have led to a dual kingship, as it was already made before (think Edgar/Knut). As in two kings of England, each one ruling his own (there was a discussion some time ago about it, my take is that the separation would have been somewhere in Mercia or along Wessex lines).
Giving that Anglo-Saxon forces wouldn't be beaten only by Stamford Bridge's defeat, unlike Hastings that cut Godwinssons and other pretenders from the better holdings of the kingdom, William would still have to deal with at least Wessex and Sussex conquest.
Granted it would be easier than IOTL, meaning William would have more fresh and more well revitailed troops than Harald.
Eventually, I think both would keep their word (for the time being, after that, you'd end with another war of unification) because the gambit wouldn't be worth the possible losses.
As for Harald being warmly welcomed by Anglo-Saxons this is highly implausible. Granted, Scandinavian have a long history in England by this point. But it should be remembered that this history definitely didn't end well for them, and that one of the decisive points of the newborn English identity was "not Scandinavian" (not only since Alfred the Great, but as well after the struggles of the XIth century). They simply let a bad memory and no one would be really thrilled to see them back in charge (Totsig revolt can be explained less by a love of Harald than being shunned by other Anglo-Saxons lords and doing a great "take that" to them)
At best, the Anglo-Saxon nobility would try to play the Norwegians against Normans (that, eventually, had as well a claim of common history with England, seeing that the massive part of Scandinavian settlers of Normandy were probably Anglo-Scandinavians).
As for Harald's dominion over his share of England, I simply don't see it lasting.
Scandinavian kings never really managed to get a lasting hold on England, mostly because they had to deal with multiple rebelling realms (England, Danemark mostly), and a Norwegian England that doesn't include the most populated and wealthy regions of the isle have its days counted.