The standard explanation, which you will pick up from any textbook on the subject, is that English came to the British Isles in the 5th C, based mostly on the language of the Angles, but including the very similar languages of the Saxons, Jutes, and Frisii, all of whom were invading - or migrating, being invited as mercenaries, trading from temporary quarters; the usual mix at the time - starting in the 400's, responding to the collapse of the Western Roman Empire in the north and subsequent power vacuum. As a side note, you will find all sorts of dates for the collapse of the empire. Closer to Rome it looked more like a transfer of power to a new tribes with only slow effect. But in the north, especially Britain, the deterioration of order and the collapse of the economies was visible within a single lifetime in the 5th C.
There are now those who claim that some version of Proto-English, or at least some West Germanic relative, was spoken in England before that. The Belgae, who were prominent in England when Caesar arrived in 55 BC, claimed to have Germanic ancestry. That may have also been a "we're really warlike" claim. So...maybe spoke a Germanic language? Though there is no positive evidence for that language part.
It would not be surprising, as tribes moved back and forth across the channel for thousands of years before that. Moving by water was still easier than by land beyond a few miles. We can now see a lot of genetic sharing in the area, enough that the commercial DNA companies cannot well differentiate between East Anglian and Low Countries ancestry. My wife's DNA shows nothing from the Netherlands, though her mother was conceived there, of two Dutch parents whole lines we can trace further back. My DNA profile shows only a little East Anglian, assigning my SNPs to many nearby places in England. Yet I can trace plenty of my Mayflower side back into East Anglia and then a few centuries further, just there. It isn't ridiculous. There just isn't that much difference among them more than 1000 years back. Who moved in and who moved out, who got overrun and who did the overrunning, who have traits gradually fall out of the pool - the similarity is greater than the difference.
Yet possibility that a thing could happen is not evidence that it did. There is little written evidence, and that from outsiders. This is before even the runic alphabet. Place-names have often been looked to to offer some evidence. The controversial Theo Venneman builds his theory of finding hints of a pre-Brittonic language he calls Vasconic from European toponyms. I am told that this practice is applied more generally for more conventional reasons by other historical linguists. Toponyms are often very old words in a place, and even when they have the appearance of the prevailing modern language, can be shown to be based on an earlier word that got repurposed from the old language to the new. That "avon" means "river" is an illustration of that. Not to mention Danube.
Here's a fun clip from the BBC about a decade ago.
We would need other evidence, or a good deal more of this. The DNA split in the eastern vs western halves of England is well-known, and as the idea grows more solid that the 5th C and onward "Anglo-Saxon" invasion was more a replacement of elites than a wholesale population replacement, it would provide a plausible explanation for the existence of something like English before that time. Still, it's not actual evidence, only an undermining of some of the previous theories.
The books mentioned near the end, Stephen Oppenheimer's The Origins of the British and M J Harper's The History of Britain Revealed: The Shocking Truth About the English Language are still quite available. Harper's is reportedly funny but insulting to the stodgy academics who refuse to accept his findings. Oppenheimer is more respectable and probably more reliable. We all like the idea of the maverick who puts the other experts to shame and eventually wins the day, and from what I can see, Harper is right, at least about the generations of linguists before this one, that the experts are a pissy, vindictive bunch. The name Noam Chomsky comes to mind, for example. But pointing out that they probably would reject someone with a new theory even if it were correct doesn't provide any evidence that you and your theory in particular are correct. It is an obvious truth when one states it that way, yet it is frequently overlooked in popular culture. No, I lied. It is always overlooked in popular culture, and the guy or gal with the new explanation, shouted down by the toffs and bucking the odds, is automatically believed by an unfortunate percentage of the populace.
When the Romans arrived in 55 BC Tacitus doesn't describe the language as more like the Germanic peoples than the others Celtic ones about, as Oppenheimer claims. (Though I have heard that claim elsewhere and don't know the Tacitus passage in question, so be wary of my claim there.)
Anyway, fun stuff, and interesting where it goes from here. As it is more than a decade beyond the information I give here, there may be news that hasn't reached me out here in the colonies.