I've recently been doing some reading into the 'Heptarchy' (although I'm aware that many don't consider this a proper term) of England and the Anglo-Saxon kingdoms. I was wondering whether the Anglo-Saxons completely wiped out the Celts, the original people of the area, (which seems unlikely) or whether the Celts were still the majority but just a subservient one. If the latter is the case, I'd be interested to know about whether they were particularly important at the time within England and whether the modern population of England is more likely to have Celtic or Anglo-Saxon heritage.
Firstly, it's important to understand that there was no single 'Celtic' group in the British Isles. Celtic populations living in different areas would have been quite distinct from each other, they may all have shared cultural commonalities such as speaking Celtic languages, but the Celtic populations were themselves present as a result of migration waves originating in different areas and taking different routes and their genetic make-up would have reflected that.
For more on who the Celts were see /u/Daeres excellent answer in this previous thread
Genetic analysis of modern populations, controlled to account for more recent migrations, shows that the Anglo-Saxons and later Norse Vikings, intermarried with rather than displaced the Celtic populations they found.
In some areas of the South East of England the proportion of Anglo-Saxon ancestry is recorded at up to 40%, but even in these areas it does not dominate.
With all that said, the genetic contribution of the Anglo-Saxons is very high when compared to other groups one might expect to have made an impact such as the Romans, the Danish Vikings or the Normans.
http://www.wellcome.ac.uk/News/Media-office/Press-releases/2015/WTP058941.htm
If I were to answer this question in a sentence, it would be that the Germanic immigrants and original Romano-British population blended together through intermarriage and became the later Anglo-Saxons, a new people group whose heritage can't be traced to a single origin.
How precisely this process went down has, however, been the subject of a long debate that's far from resolved.
The traditional story is that the Angles, Saxons, and Jutes invaded England in the mid-5th century and drove out the Romano-British violently. This story was compiled in the 1870s by a group of antiquarians and local history buffs in Oxfordshire, and it became the narrative used by British history textbooks for the next century (Sam Lucy has a chapter on this, if you want to read more).
This story has problems, however. It's based on a few different texts, none of which are especially reliable. The first is Gildas, a 6th century Welsh preacher who explains in his only surviving sermon that God will bless his listeners if they repent of their sins, just like he blessed their ancestors who won a great victory against the Saxons conquerors who burned down most of Britain in the late 5th century. I talked about some of the reasons we shouldn't trust this text last week, but it boils down to the fact that he seems to be playing pretty fast and loose with his historical details, because he's not actually interested in giving us an accurate history; he's interested in convincing us (his listeners) to repent of our sins.
However, Bede (writing in the early 8th century) relied pretty heavily on Gildas when writing his religious history of the English people. Bede, too, was more interested in the theological significance of the history he was telling than he was in presenting a strict, objective account of events. And it's clear, looking at him closely (as Guy Halsall recently did), that he fudged some of the important details about the Anglo-Saxon invasion.
After this, we have other, even later texts like the Anglo-Saxon Chronicle and Nennius' Chronicle, both of which were written to make sense of contemporary politics (through the lense of history), and both of which are written about 400 years after the supposed invasions. This is where a lot of the detail of the Saxons and Britains fighting for control of England comes from (including some of our first solid mentions of King Arthur's battles to stop the Saxon invaders, recorded by the Welsh author Nennius), and it's very difficult to rely on any of the details as more than 9th century historical mythology.
However, early 20th century historians and archaeologists did generally trust these sources, and a number of books were written that attempted to fit archaeological discoveries into the mythological frameworks of Bede and the Anglo-Saxon Chronicle. It was noted that cemeteries in England, the areas supposedly conquered by the Angles and the Saxons, had different burial practices than those in Wales (where the Romano-British were supposedly driven in retreat): 'Saxon' women were buried with disc brooches, 'Angle' women with cruciform brooches, and both tribes' men with spears, shields, and swords (which were doubtless the very weapons they used to fight King Arthur's men). Recent archaeologists have pointed out the circularity of these arguments: historical texts are 'proven' by archaeological discoveries, and those archaeological discoveries are interpreted using information from the texts. Or to put it a different way, we know that these new graves with weapons belong to Saxons because Bede says there were Saxons here, and we know Bede is an accurate source because we found these Saxon weapon graves!
Circular or not, these arguments allowed archaeologists to explain different 'groups' of artistic styles and burial practices as being the same as cultural / political boundaries, and you get maps like this which show archaeological distribution patterns superimposed with the names of the tribes thought to be associated with each group. This approach to archaeology is called 'culture history,' and reached its height in Britain and America in the 1950s.
That's the background of the traditional invasion story, in which the Angles and Saxons drive the Romano-British into Wales.
After the 1960s, Anglo-Saxon archaeologists started to pay more attention to the economic systems of the people they studied (rather than just trying to map the cultural areas of people described in texts). As they did so, they started to question the narrative of invasion from several angles. The first was that there wasn't evidence for the kind of violence texts like Gildas described: he says that the Saxons burned down all the Roman towns, but excavations in the 20th century revealed that Roman towns were actually gradually abandoned, over a period of several generations between c. 475-525. Villas, also, experienced a few generations of decline before being abandoned, and Britain seemed to enter a period of economic collapse before gradually recovering at the end of the fifth century.
Archaeologists began, therefore, to question the extent and significance of Germanic immigrants. Fifth century evidence suggested that these immigrants arrived slowly, and only took control of England gradually as they helped to rebuild its ruined post-Roman economy. The collapse was connected with Rome's abandonment of the island rather than war or invasion. And many county- and site-level studies began to suggest that the Romano-British population played an important role in the recovery alongside the Germanic immigrants (noting, for example, that the early kings of 'Saxon' Wessex have British, not Saxon, names).
By the end of the 1980s, most British archaeologists had rejected the historical accounts of invasion, and argued that the archaeological evidence showed small numbers of immigrants taking control of economic systems that relied still heavily on Romano-British peoples. Some even suggested that Britain had reverted to a tribal, Iron Age style society similar to what had existed before the Romans arrived.
That's the archaeological revisionism of the 70s and 80s, which directly challenged the traditional views drawn from (unreliable) medieval histories.
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