That depends on which theory of the Advent of the Saxons you believe in.
There is the conventional theory supported by early writers like Gildas and Bede that the Anglo-Saxons migrated during the late fifth and early sixth centuries taking over Eastern Britain massacring or assimilating the native Britons whilst driving others into the west. Historically it has been assumed that there was a mass folk migration which left large areas of western Germany denuded of population.
More recently DNA testing has been said to show that the Anglo-Saxon migration was relatively limited, that only a relatively small "elite" migration took place with a warrior class from Germany taking over the eastern counties with the local population more or less remaining in place. Said DNA tests seem to show that the vast majority of people still living in these areas are descended from the hunter-gatherers who migrated northwards following the retreating ice sheets of the last ice age. Assimilation then, not annihilation.
However this beggars the question why are there so few words of the English Language - aside from the names of a few landmarks - that can be shown to be derived from “Celtic” originals? In other instances of an “elite takeover” from our history there has been left some kind of linguistic fingerprint. The classic example being the Norman Conquest and the number of English words of French origin that can be directed linked to 1066.
Recent studies of linguistic drift seem to indicate that the point where English diverged from other Germanic languages was much earlier than the traditional date of the fifth century. This has raised the interesting possibility that the languages of eastern Britain pre Roman invasion might have been more closely related to continental Germanic than those spoken in western Britain and Ireland.
Those DNA studies mentioned above also seem to show that migration into post Ice Age Western Europe followed 2 routes. One group moving out of the “Iberian Ice Age Retreat” followed the coastal route passing through western France, Brittany and into Cornwall, Ireland and Wales. The other route was from the “Balkans Ice Age Retreat” which followed first the Danube and then the Rhine ending up in the rich tundra on the bed of the now North Sea before it’s inundation forced a retreat to higher grounds to the west. This hypothesis has it at that pre-Roman Britain had a cultural and linguistic divide running roughly down the Pennines which form the geographic backbone of Britain.
Proponents of this theory tend to point out that classical writers tend to place tribes they call Celtic in western France, no further north than the rivers Seine and Loire. North of these rivers place name and tribal name evidence seems to indicate that the Gaulish tribes there spoke Germanic dialects and we know from the accounts of Jullius Caesar and other Roman writers that some of these tribes had close cousins on the opposite side of the Channel.
Although the jury is still out on this “Doggerland Theory” my personal opinion is that prior to the growth of organised nation states rivers and seas were not barriers but avenues for the movement of goods and peoples. That for thousands of years after the formation of the English Channel and North Seas that peoples moving west on reaching the sea and being conscious of rich lands beyond didn’t stop moving but instead got down to build themselves boats.
And if when they arrived safely they found themselves on some marginal land unclaimed by any earlier settlers no one would be bothered if they set up house there. In the days of the Roman Empire if these newcomers didn’t cause any problems the first impulse of the authorities would be to tax them!
It was only later as the Western Roman Empire collapsed that parties of barbarian Germanic adventurers trying their luck in Britain were able to take over large parts of eastern Britain where the locals spoke an intelligble dialect of the language they were speaking. This allowed an easier assumption of authority and make it more natural for the natives to identify with their new ruling cast.