To answer this question, I think it's worth getting into the historiography surrounding the issue. For a long time--pretty much since the eighth century, in fact, when Bede wrote that the English were descended from the Angles, Saxons, and Jutes--it was held that the Anglo-Saxon takeover of Britain was a complete extermination. In this view, the Romano-Britons who weren't killed were pushed to the extremities of the British Isles or Brittany. Then, in the second half of the twentieth century, the discussion changed. Scholars began to gradually decrease the number of migrants until the picture was of a warrior elite who replaced the native rulers but did not replace the overall population, and acculturated them instead.

Both of these theories have serious problems. In the first scenario, such a total wipeout of the native population is, to put it mildly, pretty unlikely. The Anglo-Saxons weren't particularly technologically advanced, after all. It doesn't help that in the eighteenth and nineteenth centuries, this view was often used to justify notions of English racial superiority over Celtic-speaking groups such as the Irish, the Welsh, and the Highland Scots.

However, the elite replacement theory is also problematic. Without any substantial demographic change, language shift is often a long, drawn-out process.* Even under the might of the Roman Empire, it took over six hundred years for Gaulish to die out (and possibly more, in isolated areas such as the Massif Central). We can also compare the spread of the Scots language across lowland Scotland, a process that involved both migration and language shift over a relatively unchanged population, but one that still took some four hundred yeats. By contrast, the Celtic language scholar Kenneth Jackson posited that Old English was dominant in Lincolnshire, East Anglia, Kent, and Sussex by around the year 550, and most of the rest of England a not too long afterward. That little more than an elite replacement could have affected such a shift in a single a century certainly seems like a stretch. Furthermore, this theory completely fails to explain why Cornwall remained Celtic-speaking for so much longer than other parts of the country. Perhaps due in part to these reasons, in the past ten years scholars seem to be moving away from this notion.

So what actually happened? In eastern England, at least, it seems that large-scale migration and demographic shift is the only reasonable answer (this was borne out by two DNA tests done in 2016 on Anglo-Saxon-era burials in Cambridgeshire and Co. Durham). Further west, there was probably more mixing, while in areas like Devon, the Welsh marches, and Cumbria, a relatively minor Anglo-Saxon contribution, more akin to an elite replacement, might well be the answer. Of course nowadays this has all gotten muddled up due to internal migration, which is one reason why tests done on modern DNA have tended to produce such inconsistent results.

To answer the specific question about evidence of Britons in the Anglo-Saxon period, in the eighth century Life of Guthlac there is some indication that a community of them might have remained living in the Fens (although the way in which this is presented in the story is very ambiguous). Another way to tell is Brittonic place names. These are nearly nonexistent in eastern England but gradually increase in frequency as one moves westward (in Cornwall, of course, they are a strong majority). Meanwhile, we can also point to the "Wal-" prefix in Anglo-Saxon place names like Walton, Wallasey, and Walsall (which mean "Welshman's enclosure," "Welshman's island," and "valley of the Welshmen," respectively) to give an indication of later survivals of Brittonic-speaking communities.

*I'll note that it has happened more quickly in some situations; however, these nearly always involve some kind of crackdown policy on a regional minority language from a strong, centralized national government. This kind of thing became much more common in the nineteenth century when ethnonationalism was more of a "thing." It is extremely unlikely that anything approaching this happened in Anglo-Saxon England.