Hoo boy!
So, how is "The Confessor" going to react to the knowledge his wife is pregnant?
Are the Godwins powerful enough to protect Aedgyth, by putting out that the child of course must be the King's, because anyone who says it has another father is obviously a iiar and traitor?
If she has a son, and said son survives to 1066--then Edward's OTL efforts to pitch the throne over to his Norman cronies would be far more awkward. On the other hand, Harold would have no claim to the throne either. Whatever the Godwins say, they'd be privately certain that Edward is not the father of Aedgyth's child and all their scheming for power amounted to putting whoever that was (and I suppose everyone would do the math, even people who haven't been plugged into the Northumbrian rumor mill) into the Wessex royal family tree by the back door, with themselves only allied to it by the distaff side.
And what would Osbjorn be doing? Assuming the King plays along with the charade the best thing he could do for his illegitimate son would be to stay quiet and let him succeed to the throne.
Could Aedgyth have the leverage to name Osbjorn the regent? Other than she and he (and his Northumbrian faction of course) I don't think anyone else would want that, certainly not the Godwins.
Now if Aedgyth's child is a daughter instead then she's in a worse position overall; she can't expect to become the mother of a king, only at best the mother of someone to be married off to someone to make him a king. Edward will have the perfect candidate in mind--the first eligible bachelor in the Norman line of succession--William Rufus, perhaps, or his eldest son if Rufus is already married.
But that's the best case. What if Edward goes ballistic at this latest Godwinite betrayal? He might go so far as to have Aedgyth killed before she can even give birth! An unusually bold move on his part but maybe that last straw really makes him snap. It would amount to suicide of course, with both the Godwins and the Northumbrians out for his blood. Maybe he'd arrange for a whole lot of Norman supporters to ship over to the Kinglands before giving Aedgyth the axe, and face down his earls from behind their ranks? A different twist on the Norman invasion--instead of having to launch a dubious seaborne invasion against a pretty strong King Harold, William comes in force by invitation of the sitting King, the last of the old Wessex line, who hands the succession over to him!
But of course only over the dead bodies of the Godwins and all the force they can muster. Which might be considerable. It's bad PR for them that their daughter couldn't be bothered to stay in the bed they'd put her in (we aren't talking the morality of the freedom-respecting Northerners here but the more sexist Southerners after all) but the chips are down, Edward has betrayed the nation to the schemes of a bunch of Normans; England will fight.
Perhaps Aedgyth makes her escape. What to do with her? Supposing that the English Earls prevail against their traitor King and his imported henchmen, that none of them cut a deal with William, I guess what they'd do is crown Godwin King, with Harold his obvious heir and zealous lieutenant. Then marry Aedgyth off to Osbjorn if he'll have her, and their child is a union of the Northumbrian family with the powerful Godwins.
Or in another version--Edward keeps up the pretense it is his child, and if it is a son we have in reality and by gossip a new English dynasty, again a union of the Northern lords with the Southern, largely under the latter's influence of course. The widow Aedgyth can hardly marry one of her own brothers, so there is no logical course open but for her either to live out her life as a chaste widow--or given her character as recently demonstrated, marry her off to...well, why not a dashing if scarred heir to the lordship of Northumbria? He'd have to come down to London where "Edward's" child is being raised to the kingship, with his Godwin mother no doubt his regent.
In that scenario the Normans are going to find their claim via Edward's favor gone, and they might simply try to invade on their own hook without any pretenses to legitimacy save their own might.
That could go either way--but I suspect one purpose of this timeline is, no matter what happens to the South of England, Northumbria remains free of Norman rule for all time--even if William can subdue south England, the resistance flees north to face an unbeatable Osbjorn, who may or may not have Aedgyth by his side, and whose successors would then claim all England via her southern lineage.