This is still contested, and a lot has been said on the matter. One could look, for instance, at Henry II, who was the first English king since William I to give his eldest son England instead of Normandy. On the other hand, many chronicles continued listing the Norman/Angevin rulers as Dukes of Normandy first, then as king of England (though this often depends on where the chronicle was written). Further still, one might argue that England was still an afterthought to the royal family all the way through when King John lost most of his family's continental holdings, since his brother Richard I had effectively used it as a cash cow to fund his crusading venture and his predecessors had all spent most of their reigns not in England.

There was never any question of his practical right to rule. He showed up with an army and beat the other contenders, therefore he gets the big chair. The debate was always on the more intellectual/theological/philosophical/genealogical right to rule. There were numerous rebellions, and William put them down hard. How long did it take the nobles (which includes bishops and other men of the cloth)? Not long at all if they wanted to keep their stuff. How long did it take the common man? Much more difficult to say because they didn't keep many records, and when they appear in the chronicles it's often due to some personal obligation. The best answer may be that for small, unimportant farmers (i.e. most of England's population), there was no change other than the guy who ultimately benefited from your tax payments changed - unless your regional lord rebelled, in which case you should hide in the woods.

There's a whole lot more that can be said on the matter, but I'll just get to some of the books.

See, for example

Marjorie Chibnall. The Debate on the Norman Conquest.

Lynn Stanley. The Island Garden: England's Language of Nation from Gildas to Marvell. (Notre Dame, 2012)

D. Ibetson. Common Law & the 'Ius Commune' (Selden Society, 2001)

T. Penster & D.L. Smail. Fama: The Politics of Talk and Reputation in Medieval Europe (Cornell, 2003)

Elaine Treharne. Living through the Conquest: The Politics of Early England (Oxford, 2002)

Ann Williams. The English & the Norman Conquest (Boydell, 1997)

My current research deals with questions like these, so my list of books directly or peripherally related is well over 500 now, so if you want anything more specific, let me know.

Hope this helps a little, happy reading!