Increasingly, the evidence points toward peaceful intermarriage between the Romano-British (Britain's 'celtic' people) and the continental newcomers.

Our one contemporary textual source, a Latin sermon by a Romano-Britain named Gildas, talks about violent conquest by the Saxons who drove the Romano-British 'celtic' peoples of England to the west, but there's good reason to read him with a great deal of critical skepticism. I've posted about this before but, in short, it boils down to the fact that Gildas is clearly playing fast and loose with historical facts in order to create punchy sermon illustrations that support his claim that God hates bad kings.

Our other textual sources that talk about violent Saxon invasions that drove out the Britains (Bede, the Anglo-Saxon Chronicle) relied heavily on Gildas.

Archaeological evidence suggests a much less violent picture. People from Germany show up in the Roman army in the fourth century (if not earlier), and more continue to immigrate after Britain leaves the empire in the fifth century. These early immigrants lived on farms, appear to have been pretty poor, and intermarried with the local population. Many older texts will point to the weapons buried in some of these 'Germanic' graves as evidence that they were violent warbands, but in fact graves with weapons didn't become common until a full generation (or two) after the year our unreliable texts say the violent invasion happened.

And increasing amounts of evidence are showing that these newcomers mixed with the local population, rather than replacing it. One method that's become very useful over the past decade is to study to ratios of stable isotopes in people's teeth. These isotopes reveal a person's diet and place of birth. And the pattern that's emerged from these studies is that people buried together in the same small cemetery (often about 200 bodies from 150+/-50 years of use, meaning 5 or 6 households) is that some of them came from the continent, some came from western britain, and some grew up in a neighboring area several miles away where the groundwater was different. Ie, people moved around a lot and intermarried; the Germanic newcomers didn't drive the Romano-British people off the land, they moved in alongside them.

DNA studies are starting to produce interesting results as well. A recent study published in Nature found that Britain's (modern) DNA, even in the part of the island closest to the North Sea where the invading Saxons were supposed to have driven out all the locals (if you believe, say, Härke's arguments), was mostly descended from the neolithic migrants who settled Britain thousands of years before the middle ages (a German genetic haplotype shows up in the modern inhabitants of southeast England, but it suggests only something like 10-40% of the population in that region were immigrants - the rest were the locals they settled among). And studies of DNA in the bones of early medieval people from individual cemeteries are just starting (using really new techniques) to give interesting results - there was a paper at the SMA conference last week about a cemetery excavated in 2012-14 where the DNA sequences showed an even proportion of continental, Romano-British, and mixed ancestry. And the isotope study of the teeth in that cemetery showed that they all ate the same food - ie, they were part of the same community in life and maybe even ate at the same table; they were a community, not richly fed Saxon masters and their poor British slaves.

Saxon / Germanic culture won out in the end as elites decided they would rather focus on the German side of their ancestry instead of their Roman past. But this was more a political decision than an ethnic necessity - England's Anglo-Saxon identity was created by elites during the social, economic, and political changes of the sixth and seventh centuries; the same processes that led other parts of Britain, like Wales, to choose to emphasize their Roman roots. If they'd wanted to, however, the people in southeast England could have chosen to go the other way - they had enough British ancestry to make an honest claim to be inheritors of Roman Britain. But, by the seventh century, being Roman was less useful than being Germanic (you see the same shift toward a unified Germanic, 'Frankish' identity in neighboring Gaul in the early seventh century), so the celts were written out of English history (mostly - some Celtic names survive in the geneologies of the kings of Wessex, for example), towns were renamed with English names, and historians like Bede began to create the official story of England's exclusive descent from the Angles, Saxons, and Jutes.

But the reality was: the Anglo-Saxons were as much, or more, British and Roman than they were Germanic. But by the eighth century, they would probably have denied it.

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If you want to read more, I would recommend James Gerrard, the Ruin of Roman Britain. He provides a good discussion of the archaeological evidence and frames it within an excellent narrative of Britain's transition from Roman to Anglo-Saxon.

Robin Fleming, Britain after Rome, is also excellent. And it's very readable (more so than Gerrard), inexpensive, and nevertheless thoroughly researched by an excellent scholar.

Guy Halsall, Worlds of Arthur, has a great discussion of the surviving evidence from the period, and provides thorough take-downs of the many ways this evidence has been misinterpreted and abused by historians and archaeologists over the past century. If you want to know why not to believe the story you've been told about the Saxon invasion, he's the place to start. His own alternative interpretations at the end of the book are thought-provoking, if tenuous.