I think the question would be, what makes someone "Anglo-Saxon"?

If, by this question, you are asking, "did immigrants ever outnumber people born locally," the answer is almost certainly no. There was a recent study of (modern) English DNA, and it was able to identify very small variations in people's genes that suggest that many people in southeast England shared ancestors with people living in modern N. Germany about 1200 years ago. That study suggested that if this genetic similarity resulted from a mass-migration of people from Germany to England in the early middle ages, these migrants would have had to be between 10-40% of England's population, ie never the majority. There is much other archaeological evidence that indicates intermarriage / mixing between migrants in the early middle ages and local people, so this study was not surprising. Many people migrated, but never so much that they fully replaced the local population.

However, a recent followup to that study (by one of the same authors) questioned whether the modern DNA evidence resulted from one single migration event. It could, actually, be better explained by two separate migrations--one, the Angles and Saxons, and a second when the Great Heathen Army (ie, the Vikings) invaded in the 9th century. I spoke to one of the co-authors of this study, and asked her if there could be three, or four, or dozens of smaller migrations--and she said absolutely yes. If there were smaller migrations, this would mean that the proportion of immigrant people present in England at any one time would be smaller, compared to the locally born population.

In fact, other archaeological evidence supports the idea that migration was continual, not a one-off (or twice-off) event. You can, for example, identify where someone was born by their teeth. The water we drink in our childhood lays down isotopes in our enamel, and archaeological scientists can extract these isotopes and identify where someone grew up. These isotopic studies show the migration was continuous during the early middle ages--immigrants from Germany and also from the Mediterranean and even Africa appear in cemeteries from basically every century we have studied. As further confirmation, a recent study of the DNA of several 6th and 8th century skeletons found evidence that some of them in both centuries were probably immigrants from the Netherlands (again, looking at very minor genetic mutations that tend to cluster in geographic regions).

So if the question is, did immigrants ever outnumber locals?, the answer appears to be very firmly no, given the evidence currently available from the archaeology.

However, it gets more complicated than this if we change the question and ask, "when did Anglo-Saxons outnumber Romano-British persons?"

Anglo-Saxon is an ethnic identity, and ethnic identities are not like genetics. Your genetics depend on your ancestry, but ethnic identity is about which group you belong to. There are many ways to belong. People can assimilate to an ethnic group by changing the way they dress, learning a new language, or adopting new cultural practices or foodways. Similarly, ethnic groups can change or even be created from scratch. "Anglo-Saxon" is a great example of this, because there was no ancient "Anglo-Saxon" group. The name "Anglo-Saxon" is a combination of the names of two different tribes, and it only shows up in later written sources. There is no evidence that a man or woman in 6th century Suffolk would have identified themselves as "Anglo-Saxon" (instead, they probably would have said, "I'm one of the South Folk"; that's where the word "Suffolk" comes from). But over time, "Anglo-Saxon became a catch-all term for all the peoples in England, as a way of distinguishing themselves from neighboring kingdoms like those in Wales, Scotland, or across the English Channel / North Sea.

So we could ask, when did southeast Britain develop a culture that was uniquely different from that of Wales, Scotland, etc? There are several answers we could give. We could say the fifth century, because in the fifth century Roman institutions in the southeast of Britain collapsed, hard, while those in the West and North continued to limp onward with more continuity (perhaps because the West and North were never as fully Roman, and consequently more able to survive the collapse of Roman infrastructure, while the more thoroughly Romanized Southeast fell to pieces when Roman infrastructure faltered). You could also say the late sixth century, as this is when strong new kingdoms assert themselves in the Southeast--the kingdoms that would, in later texts like Bede's history or the Anglo-Saxon Chronicle be identified as the kingdoms of the Angles, Saxons, Jutes, etc (though genetically, their people were far more diverse). This is the period where English became widespread--probably. You could even push it to the tenth century, when Athelstan finally united all these kingdoms under the rule of Wessex, creating an "Anglo-Saxon" kingdom across the whole region that we think of as English for the first time.

One could, however, draw the division much earlier. The parts of Britain we call "Anglo-Saxon" correspond with a geological region called the lowlands. This region has a different ecology than the uplands. Archaeologically, there's a sharp line between these regions all the way back to the Bronze Age (and probably much earlier). Several recent studies have suggested that "Anglo-Saxon" England was shaped more by this geological division than by the culture of immigrants. People in the lowlands intermarried, shared culture, and developed in one direction, while people in the uplands did their own thing.

Regardless of which model you prefer, the answer to the cultural question depends less upon the number of migrants, and more upon how people chose to live their lives. People in the parts of Britain that, certainly by the tenth century, were calling themselves "Anglo-Saxon" shared many aspects of language and culture that were different from Wales and Scotland. These differences did not come from a shared immigrant experience--most of these people were born locally, and descended from people who had immigrated in prehistory with some later immigrants mixed in. Yet culturally, they adopted a language shared with Frisia across the North Sea, as well as material objects linked with Germany and the migrants who did cross the North Sea in the early middle ages. These changes were not a result of overwhelming demographics: they were the result of people's choices. They chose to emphasize an immigrant identity that was, for most, not the biggest source of their ancestry. For these people, however, it seems to have been the most important part of their family histories.

So at what point in Britain's history did Anglo-Saxons outnumber the native British? When the descendants of local British people decided to identify with their Continental ancestry instead of their local connections. As far as we can tell, that shift happened, probably gradually, between the sixth and tenth centuries.

Note finally that the pre-medieval inhabitants of Britain were not a single group of "Celts." No one in Wales of Scotland called themselves Celtic, and their languages were not by the early middle ages mutually intelligible. There were many different regional groups of people in the non-Anglo-Saxon parts of Britain, and they--like their neighbors--also changed how they expressed their identities over the course of the early middle ages.

(Sources / for further reading available on request.)

Hello, I sort of answered that question here and the answer is that really, they might never have. The old school of thought, that a mass migration of Anglo-Saxons displaced the native Britons, has been largely disproven and it seems far more the case that Romano-British society was assimilated.