Recently, we have heard some persons claim that the Normans who invaded England were almost all Celts, Duke William included, because this was who populated Normandy. This belief requires a great unfamiliarity with northern history and nomenclature.
Perhaps too much so, lately it has become acceptable to refer to the Norse of the Migration Age as Vikings. This has helped this confusion. It might be better to revert to the less particular and more descriptive names for these peoples: the Norse, Norsemen, Northmen, and/or Normans.
When Hrolf Ganger made his pact with the King of the Franks, his bargain was that he would become Christian and settle his men and as many other Normans as wished to join him in the lands at the mouth of the Seine as a warrior-barrier against other raiders like himself. Certainly there were plenty of live bodies in the country, Celtic bodies, but they were the disarmed and ineffectual peasantry. They could not make the least resistance to any armed force. One thing to remember: the kings and nobles liked it that way. The Franks had spent many centuries reducing the warriorhood of the peasantry, putting it down violently, outlawing the possession of warrior's equipment, and draining off their best into the gens d'armes class, the men-at-arms.
Hrolf took the baptismal name Rollo, but most of his followers even when baptised continued to use their Norse names. Not only did the area become known as Normandy, "land of the Northmen," but the names of settlements changed in the area. There were so many Norse living in Normandy that farms, villages, and towns became known as the likes of "Toki's town" (Toqueville). The very names of the place lost their Celtic character. Also, it must be noted that Hrolf's forces not only sent home for their wives, but all their cousins and anyone else who wanted free land and free serfs to work it in return for their fighting skill.
So the Normans of Normandy were very much still Normans, Norsemen, though they acquired the French language for dealing with both their peasants and their overlords. The Normans expanded greatly in numbers, so much so that they ran out of job positions for warriors. Thus, large numbers were available to resettle when some sent back word from southern Italy that there were lands and cities ripe for new overlordship, did anyone come willing to fight and conquer.
In the meantime, other Norse had settled in northern England in similar numbers. There area became known as the Danelaw, the place where Danish (Norse) law rather than Saxon law was followed. One mark of this conquest are all the places named "-thorpe" which is a Scandinavian word for "village." A second is the infamous Yorkshire dialect, opaque to most Englishmen even in the latter 1800s because it was not based on English, but was a pidgin of Norse.
Finally, the English ruling class became heavily infiltrated with Norse blood, until finally Scandinavian kings could successfully claim the British throne. King Canute was actually a Norwegian, Knut. Again, names, of persons not places, prove the case. "Harald" or "Harold" is not a Saxon name, but a Norse one. In 1066, all three of the claimants for the English throne were Norse blood: King Harald Godwinsson, whose brother was Earl Tostig of Norway who came in the invasion train of King Harald of Norway, and Duke William, probably least Norse of the three because he was the son of a tanner's daughter, but bringing a war-train of landless Normans. Englishmen might chose to follow one or the other, but it was all Normans, all Norse, who were actually grasping for the crown. It was hired Norse house carls, not Saxon, who fell in the shield-wall at Hastings, axes against the clubs and swords of their Norman cousins.
The final out-thrust of the Norman warrior-excess was the First Crusade. Virtually all the names of the ones who stayed, of note, are Norman, Norse, from Norman holdings (the exception was Raymond of Toulouse). Once again, younger sons went out to conquer new lands and hold them. In this case, the Scandinavian expansion had thinned so that reinforcements from home did not come in the necessary numbers; the Turks invading the other direction were more determined, and the religious conflicts with their subjects were not solved. There were even major health difficulties: few children were born to the Norman upper classes and the mortality was high. Many lords left lone heiresses when they died, and often no children at all. It is only in the later 1200s that we can begin to ignore the Norman factor in the pre-national politics of the times.
So let us try to keep our references in period, and not refer to peoples, especially ruling classes, as if they were of a people long-subjugated or vanished, or yet to come. Duke William's Normans were neither Celts nor Frenchmen, any more than the Byzantine emperor of his day was either a Roman emperor or an Ottoman sultan.