It is still disputed as to which Norse nation the Normans are actually descended from: Norwegians or Danes. It is not known whether Hrolf/Rolf/Rollo, the Norse founder of the duchy of Normandy, was Norwegian or Danish. Though one thing is known: the Normans were not full-blooded descendants of Vikings but a meld of Norse and Frankish bloods, both of which are of course Germanic. Norse settlers in Normandy intermarried and interbred with Frankish women as well as adopting Old French as their language, Christianity as their religion, and the Frankish form of feudalism as their system of government. After a time and certainly by the 11th century, the Normans regarded themselves as Franks.
To further explain, the Normans are a product of Frankish/Viking culture. The emphasis is on the Frankish culure rather than the Norse, but if you consider the genetic make up of the Norman population, a very large part of it would have to come from people whose ancestors were Gaulish.
The Gauls stop being Celtic soon after Caesar conquered Gaul in the 50s BC. Other than their genetic makeup there was probably nothing left of the Celtic culture to contribute to the new Frankish culture, certainly even more true with the later Normans. The only possible except are the Britains who settled in Brittany.
I have seen maps of Norse place names along the coastline of Normandy. Certainly this is proof of Norse populations. However, when taken in context with all other placenames, I suspect Norse names are a distinct minority. Just as an aside, William’s favorite town, Caen, is an ancient Celtic name.
The "Franks", who invaded/immigrated after the Roman Empire broke up, were "Germanic" people. These people and the Romano-Celtic groups thoroughly mixed after that, and that's one reason there are problems finding any distinctive genetic material. Later, because the Norse were something of a minority in Normandy, probably not a whole lot is left of their genetic legacy. On the other hand, it's still there in places. Place names give a clue that they were there, but one can't read too much into the fact that a place like Caen has a non Scandinavian name. Because maybe people liked "Caen" better than something like "Eyvindburh" or the French equivalent.
In addition, There are generally 2 groups of Germanics: The West-Germanics and the North Germancis (Viking peoples). However, both are, somewhat, related; see common pre-Christian culture (Althing, Dieties, Sagas etc).
There must have been mixing between the Gallo-Romans and the Franks, then between the "Franks" and the "Normans".
Some mistake cultural for ethnicity. Even before the Franks mixed with the Gallo-Romans they were a confederation of various tribes. When you add up all these different ethnic groups that went into making the Normans, it's much more complicated than just calling them Vikings. I think this is one of those things that most historians gloss over, opting for the overly simplistic explanation.
As far as the DNA is concerned, the y-chromossome doesn't get mixed up in the genetic recombining that occurs with each generation. That's why it is so valuable for ancestral research. The y-chromosome of a man today is, for the most part, an exact copy of his male ancestors 100s of years ago. In SOME cases that lack of change can go back tens of thousands of years. Testing is not, yet, accurate enough to differentiate between Viking, Angle, and Saxon DNA in England.
English, Scots, Welsh and Irish are very closely connected genetically; in fact, there is very little to differentiate at all - they're much closer to each other than they are to their continental cousins. West British people (people from Wales, west of England, NW England) who take the DNA test tend be ever so slightly closer to the Irish than they do to the English.
However, Scots, Irish and Welsh have more similarities. The English are more 'multi-cultural' in their genetic make up with such succesive waves of settlers with influences from Celts, Romans, Angles, Saxons, Danes, Norman.
The Irish, Welsh, and Scots share the same genetic make up as the English, because of the huge Norman or Norse influence that they endured for centuries. There was no part of the British isles and Ireland that the Normans or Norse didn't dominate, since the Normans or Norse were very shrewd, and the men married into the local aristocracies, thus, spreading the Norman or Norse seed genetically. There is Norman or Norse ancestry to a greater or lesser degree in all the Anglo-Celtic aristocracies, peerages, and gentries throughout Britain and Ireland. Norman or Norse blood is a common element.
Also, The original population of the British Isles (prior to the Celts) plays a part.
Every country in the British isles has some history of English and Norman or Norse settlement. The Scottish lowland population is predominantly of Anglo-Saxon stock. The Anglo-Saxons delved significantly deeper into Wales than the modern border reflects. Also the immigration of Flemings to Britain was of much greater importance to Wales than elsewhere. Typical "Welsh" names such as Jenkins, Thompkins, etc, are of Flemish origin. Ireland was heavily settled - or planted - by English and lowland Scottish settlers over a period of hundreds of years. Finally, the Normans or Norse settled throughout the British Isles.
Here is anoher point to consider. It has been discovered that all the people of the British Isles are not as "Germanic" as was once originally thought.
New studies have genetically linked them to Celts from Northern Spain (such as Basques) and that this group of Celts sailed across the Bay of Biscay before the Romans arrived and constitute about 80% of British genetic makeup today. That includes the English too and that in fact the genetic difference between Scots, Irish and English is minimal at most. The studies showed that even in places where ther was supposedly strong Scandinavian or Norman influence, it made up at most 20% of genetic influence. (Except in the Orkneys and Shetlands - where they made up about 40-50% of the genetic makeup).
This is evidence that the Anglo-Saxons formed a small aristocracy that ruled and/or mingled with a large Celtic population rather than a large scale invasion and displacement by Anglo-Saxons. The same situation occured with the Normans or Norse. The question remains, "How much genetic influence did they really have?" Maybe they did not mix as much with the local populations as some might suggest and were mainly temporary settlers.
The evidence suggests that England is overwhelmingly genetically Celtic and it's culture is a rich mixture of Germanic and Celtic in equal parts with some Scandinavian and some French linguistic elements.
DNA tests suggest that the vast majority are of pre-Saxon descent.
As for names, the majority of river names in England are pre-Saxon, and the same goes for major towns, i.e. London, York, Leeds, Manchester etc.
It may well be that the names of smaller villages tended to change over time, but the Norman Conquest brought in written records which pretty much put an end to this. This would explain why they tend to be more English, because whatever Celtic name a village might have had earlier (if it even existed) would often have been replaced by an English one by the time of the Conquest (due to language replacement).
However, many place names which have an apparently English meaning might actually be of Celtic origin. An example of this phenomenon is the town of York, which in Old English is Eoforwic (which apparently means town of the bear); but looking at the earlier records this is clearly just an Anglo-Saxon corruption of York's Celtic name Eborac or Ebrauc; and Edinburgh is known as Edwinesburg in Old English. If York had never been mentioned in Roman or early British writings (as in the case of most small villages in England), then everybody would assume the name York is of English origin.
The Anglo-Saxon invasion was, essentially, no different from the Germanic invasions of the rest of the former Roman world.
I'm certain, at some point, differences will be seen in the y-chromosomes of different groups, but until that happens research is confined to intelligent guesswork.