Trying to look at the balance of forces, Godwinsson has a serious defenders' advantage; he has a mobile army that can probably do the job itself, but if he is given time in one area, he will be able to rally the local fyrd to him.
This is the key to the operations of Hastings, incidentally- Harold had enough of his household troops to contain the beach head, but not enough to squeeze it out; he wanted to hold William in place until the fyrd could rally, and William took a long shot on breaking out before that could happen.
Back to the North. With an allied Scots army in tow- competently led, but with the usual characteristic flaws of a Scots army, the terrible tendency to gamble for victory combined with the stubbornness of refusing to admit it hasn't gone well, and doubling up on a losing bet- Fulford's a stroll.
How well the Scots behave- whether they are an asset or a problem in lands that are already full of reasons to hate them- debatable, but under good officers (anachronism but you know what I mean), York and environs are taken and held.
There are so many false starts in 1066, William tried and failed to make the Channel crossing several times before actually succeeding; there's a butterfly to be had there. If not, well.
Hardrada was a wild man in his youth, but some time in Byzantine politics seasoned him; the impression I always got was that he would have been happy to winter at York, spending time on the civil reconstruction of the Danelaw, before moving for London in early spring.
Stamford was a strategic ambush; Godwinsson was not expected, which in itself is odd- perhaps the presence of the Scots changes that, simply because of the larger army having to be billeted over a larger area, more foragers out, or perhaps highland warfare has taught lessons and they have proper pickets out.
So it turns into a proper stand up fight, which the Norse-Scots Alliance wins, but then what? The army's not going to move the way Godwinsson's did, by land. That's what the drakkar are for. Faster, too.
Coming south by ship, once persuaded of the necessity, there are three things the alliance can do; head for the capital, take and hold before William can, and strike out from there;
land in the Thames estuary and March to intercept William, which could mean a meeting engagement around the lakes to the west of Ashford;
or gazump William's beach head, land at Hastings themselves and take the stores, horses and so on William's already collected for himself, and roll the Normans up from the rear.
You're going to be writing it, but I would reckon that after surviving Stamford, Hardrada's blood will be up and he will be spoiling for a fight. Yes, should be enough.