(Author's note: History is a wide ranging discipline which in some senses can be fluid as we learn new things and in some cases is solid as we choose to interpret events and findings in light of our own day, not the day in which it happened. Individuals spend their whole lives studying these things. My very concise overview is meant as nothing more than that: an overview to give background. All errors and omissions remain my own.)
Once upon a time in the not too long ago, English School children and those that derived their existence from English history learned that the Anglo-Saxons invaded the island of Britain in 449 A.D., where they effectively moved in and took over everything.
As usual, the truth is somewhat less well defined.
Map of Britain 450-475 A.D. - Source
Whether the Romano-Britons of 446 A.D. heard a response from the Emperor about the status of their land (The Groans of The Britons) or not is not recorded for us. What is recorded is that actions that they took, which suggests the response was either "Look to your own defenses" or simply no response at all. So look to their own defenses they must.
The ongoing rebellion of the already hired Anglo-Saxons meant that hiring additional Angles and Saxons might not be a good idea. The shadowy figure of Vortigern appears again when, to repel an attack of Picts to the North, he hires a different set of Germans, the Jutes (perhaps so-named from the Jutland peninsula) to assist. We have the names of these leaders, Hengist and Horsa, although we have no idea if they really existed or were mythical figures (the word "hengist" in Anglo-Saxon is one word for "horse").
Hengist and Horsa may not have been real. Their landing in Kent likely was.
It does not seem that they were allies of Vortigern for long. In 455 A.D. they are recorded as fighting with Vortigern. Hengist was allegedly slain; Vortigern also falls out of the few records we have of him, perhaps slain as well.
Map of Britain 475-500 A.D. - Source
From these humble beginnings, the true invasion of Britain commenced.
Initially the Romano-Britons had a hard go of it. Theirs initially is a history of defeats of four battles over 25 years (456 A.D to 473 A.D.) - which perhaps makes sense if we look at the wider historical record: the invaders (Angles, Saxons, Jutes) had the ability to come and go at will and seek allies as their highway, the sea, was to their backs; the Romano-British had no such ability.
And then, something of a miracles happened: There was a Romano-British resurgence.
Map of Britain 500-550 A.D. - Source
All of our written knowledge of this comes from the monk Gildas, writing in the 550's A.D. He states that a leader, Ambrosius Aurelianus, a man he calls the Last of Romans in Britain, born to a family that "had worn the purple", lead a counterstroke and heavily defeated the combined Germanic armies at a battle called Mons Badonicus. We do not know where this battle was, or specifically when it was, (although Gildas writes that even in his day, there was still peace from this battle, so the estimates run from 486 A.D. to 516 A.D), or even what the impact was from the battle, except that it halted the advance of the Angles, Saxons, and Jutes for a generation.
Ambrosius is a mystery as well: we know nothing about him, his origin, or his ultimate fate. We do know of him from legend, as his is the most likely origin story for King Arthur.
Map of Britain 550 - 575 A.D. - Source
But the tide, once started, could not be turned back. In 552 A.D., the Saxons won a victory at Old Sarum. in 556 A.D. they won again at Barbury. In 571 A.D. they won again, taking parts of Bedfordshire and Buckinghamshire. In 577 A.D. they took Glouceaster, Circenceaster, and the old Roman town of Bath. They pushed north as well, winning parts of Northumbria in 590 A.D. and engaging directly with and defeating the Scots in 603 A.D. By the time of St. Augustine's visit in 597 A.D., most of the middle and Southern part of Roman Britain had been conquered.
Map of Britain 575 - 600 A.D. - Source
What of the remaining Romano-British? Some of them fled - first likely farther west into what would become Cornwall and Wales. Some fled even further; the Byzantine historian Procopius writes from Byzantium that word of refugees fleeing from Britain to the old Roman province of Amorica reached their ears. The province, depopulated by previous invasions and plague, became the province of Brittany (and explains why a Celtic language still exists in France). The immigrants brought not only their language but their names as well: Two of the Atlantic provinces were name Cornouaille ("Cornwall") and Domnonea (Devon), reflecting very directly where they had come from.
But for the Romano-Britons, this would begin their long, slow retreat the Celtic Fringes of Europe. The future of Britain, or as it was coming to be known, Engle-land, would belong to the invaders, who would have little understanding of the history that had come before them, coming to believe that giants had built the great ruins they wandered among; it was impossible that men could have done such a thing.
Blair, Peter Hunter: Roman Britain and Early England 55 B.C. - A.D. 871. WW Norton and Company: London, 1991.
Nicolle, David: Arthur And The Anglo-Saxon Wars. Osprey Publishing: Hong Kong, 1984
Alcock, Leslie: Arthur's Britain. Penguin: Great Britain, 1971
Wikipedia: Amorica, Groans of the Britons