I don't think Stamford Bridge would be a really good PoD on this regard.
Not only Anglo-Saxon identity pretty much evolved towards a "definitely not Norse" stance, but precedent mix of population was broken (or at least made irrelevant) one century before.
Giving the (mis)fortunes of Scandinavian takeover of an already English kingship, it would most look as a personal union without that much cultural influence and probably short-lived.
If we're asking about a Norse takeover in the Xth century (while, IMO, it would have been really hard to have more than they obtained IOTL), it's easier.
It would probably be similar to the linguistical impact of French after the Norman conquest, maybe less giving that IOTL Anglo-Norman was carried out by a specific social model while it doesn't seem that IOTL Anglo-Scandinavian culture was : from what we know, a mix occured in spite of the relative short time length.
Because of this, I'd tend to be inclined to argue in favour of a bipolarisation of Old English : more norse influenced on North-West, less so on South-West.