Mmm...Depending of the context it could be really different.
First, a minor issue is that Normandy, contrary to England, Scotland or Ireland never was a kingdom to begin with : I doubt it would be called "United Kingdom".
More seriously, a more important problem is the great difference between Normandy and England : customs, language, institutions, settlement...Normandy lords tended. Arguably, Scotland was as well but England never had to dispute the control of its northern neighbour to a rival power.
Except if England holds other holding like OTL and manage to keep them (not only it would be quite hard in first place, but you would more likely end with a dual kingdom centered on Aquitaine and England rather than Normandy), odds are against them.
And if you think about "common law for everyone!", It's going to please Norman lords and elite so much that they would have made fire and arm themselves in order to show their joy.
You see, if Normandy and England were separated after William's death, it wasn't because it looked prettier on maps : managing to keep both lands and their nobility on control was better this way. OTL, Normandy was somewhat neglected by English kings (except for military matters, and not always), and if the local nobility enjoyed being relativly autonomous, the kings weren't that enthousiastic about it. They tried many things to keep control : justiciar serving as viceroy, "apanage"-like features, etc.
It didn't much served.
You'll need then to stabilize Normandy, and to keep it on firm English control.
Now, if you manage to make Normandy somewhat regularly autonomous at the end of a viceroy-like feature, it would probably help.
Of course, another obstacle is the french king : he's not only the suzerain of the territory, but Capetians's hobby was to preserve and make growth any revolt they could even by participating to them. Crushing Capetians would be decivise.
By crushing, I mean more than reverse-Bouvines, I mean make them History. But not crushing too hard, or you'll return to the part "Normandy is only a province among French holdings of English kings".