Well the first thing that's important to remember is that national identities were in their infancy in the middle ages. England is one of the few exceptions to this, but there was precious little in the way of a "French" national identity, particularly in William's case as he was in fact a Norman and spoke a different kind of French than those in king's court.

The other problem is that there is very little mention of the commoners and peasants in our contemporary accounts except to describe when violence was being done to them. In the immediate aftermath of Hastings, England was wracked by repeated uprisings among the Anglo-Saxon nobility that remained and incursions from Scotland, Wales, and Denmark. Particularly in Northumbria this was devastating to the local populace. The Harrying of the North stands out in our sources because of the harsh denouncement of William by historians only a generation or so removed from him.

He is accused of what reads almost like an account of genocide today. However we should resist applying such an anachronistic label to the campaign, suffice it to say it was considered unusually brutal and thorough.

Those particular peasants had quite a particular and drastic change in material circumstances due to the Norman conquest, but it is impossible to put together a picture for the rest of England, even with the Doomsday Book as a guide (it was only completed a few decades later).

One of the more interesting events that happened following the Norman conquest was the large Anglo-Saxon influx into the Byzantine Empire and its realm of influence. The Anglo-Saxon composition of the Varangian Guard in the years after Hastings is well attested, even if many of them fell in battle against the Normans of Robert Guiscard in his attacks on the Byzantines. There are even two sources which claim that Anglo-Saxon exiles/refugees founded a new kingdom in the Black Sea and named it Nova Anglia, but this has not been supported by contemporary evidence or archaeological finds so it remains likely legendary.

Primary Sources: William of Malmesbury, Oderic Vitalis respective accounts are a good starting point and far more in depth than the Anglo-Saxon chronicle at this point in history. Anna Komnene mentions the English Varangians in her biography of her father.

A good starting point for reading up on secondary sources would be Donald Henson, "The English Elite in 1066: Gone but not Forgotten"

On top of /u/Steelcan909's answer, one of the other changes which would have affected the everyday English population in a meaningful way was the Norman overhaul of the legal system, especially sanctioned punishments and the 'Forest Laws' regarding newly-created royal forests and hunting rights therein. I discussed that and the changes here.