The generally accepted view is that Harold's Anglo-Saxons took Harald Hardrada's Norwegians by surprise, catching them dispersed and without much of their armor and heavy equipment (as the Vikings had stowed these on their ships, not expecting another battle for days/weeks). Even with these advantages, the Anglo-Saxons supposedly had a very tough time of it with the battle lasting most of the day. In fact by some claims the Anglo-Saxons lost up to 40 men trying to overcome the first Viking warrior they came across (who was holding the bridge alone), although I can certainly understand that this bridge story may have been exaggerated somewhat.
So, what if the Norwegians had received advanced warning of the Anglo-Saxon army's approach? I suppose a key part of this question must relate to what exactly happened in OTL, e.g. were the Norwegians really caught without their chainmail byrnies and so on or was this just an attempt by Scandinavian sources to explain the defeat? And also of course how much of a disadvantage it would have been to be unarmored in the first place.
Assuming that the "generally accepted" version of the tale is more or less true (i.e. the Norwegians were widely dispersed in two or more groups, and mostly unarmored and/or suboptimally equipped), I would say that Hardrada would definitely have a good chance of prevailing at Stamford Bridge. And if this happened, although the waiting Normans would of course have benefited (as they did in OTL) from sitting out the first bloody battle, perhaps Hardrada could have counted on rallying some support from the North of England? If this was feasible then I also give Hardrada a good chance against the Normans, although if this was not possible then I doubt the Norwegians and the Normans would rush towards each other so the campaign could in any case have dragged on for quite some time, with the outcome by no means certain. Thoughts?