Following the Roman withdrawal from Britain, it’s unlikely that there was any political entity representing Britain as a whole. I say unlikely, as written and archaeological sources disagree.

Written sources suggest that smaller political entities formed in the absence of Roman power. These included a range of smaller kingdoms, whose kings answered to one overking. The narrative goes that this overking, Vortigern, invited two Saxon mercenary leaders, Hengist and Horsa, to Britain, and that they later turned on him and conquered much of what is now England.

This, however, ought to be taken with a grain of salt. Our two main authors are Gildas and Bede, both monks, whose writing was primarily concerned with religion. Gildas was relatively close to the events, writing roughly a century after the end of Roman Britain. However, his work - On the Ruin of Britain - was a sermon on the religious failings of the British, portraying the Saxon invasion as divine punishment. Bede was writing centuries later, and his work - The Ecclesiastical History of the English People - was focused on the rise of Christianity in Britain, moreso than its political history. Both of these works are widely available and translated into modern English.

We would expect to have archaeological evidence to support the existence of a nation-wide political system, of local kingships, and of a violent invasion. However, this evidence isn’t there to be found.

While we do have evidence for a couple of local kingships, operating out of former Roman cities, these were - as best as archaeologists can tell - minor powers whose influence was very local. We don’t have evidence for the minting of coins during this period, or for long-distance trade. The evidence instead points to a collapse of urbanism, with the population becoming very rural in the century following Roman withdrawal from Britain. Major cities, including London, Colchester, and York, were mostly abandoned during this period. Even rural communities seem to have regressed, moving into subsistence farming without towns in which to sell ‘cash crops.’

Nor is there evidence for a violent invasion of Britain by the Saxons. There’s no major battlefields, no mass graves, no archaeological legacy of a country torn by war. Instead, we see Romano-British cemeteries develop into Saxon cemeteries in a seemingly peaceful and gradual process. Genetic evidence - to my knowledge as a historian with no scientific background - suggests that the Saxon influx into Britain was fairly limited. The historical consensus, therefore, is that the rise of Saxon Britain was primarily driven by cultural exchange, rather than by violent conquest or by mass colonisation.

Most of this is covered by general narratives of the Anglo-Saxon period. Higham & Ryan’s The Anglo-Saxon World is the most recent overview, and is both accessible and well-reviewed. My particular interest in the early period is in the development - and lack thereof - of towns, and I can recommend more focused works on that topic if you’re interested.

TL;DR: Traditionally it was thought that local kings, answering to an overking, took the Roman’s place, before being violently overthrown by the Saxons. Archaeological evidence has since suggested that the end of Roman authority saw Britain become mostly rural and ungoverned, before being colonised by the Saxons in a surprisingly peaceful process.

Well, in part it was the Saxons.

Sub-Roman Britain is one of those genuinely rare things in history: practically an actual Dark Age in terms of readily available documentary sources for the period. The narrative established in Gildas' De Excidio Britanniae... and expounded upon in Bede's Historia Ecclesiastica - that the Romans' withdrawal led to an immediate Saxon invasion and total political, cultural and demographic replacement of the Britons - has, however, been pretty significantly refuted. There's a pretty comprehensive assessment of the current discource in Oosthuizen's The Emergence of the English (2020): essentially, archaeology, toponymic studies and some genetic studies suggest that the arrival of the English into the British Isles was a long and piecemeal process that started in part before the withdrawal of Roman authority and, while in some instances it was a process of conquest, in many others it seems to have been one of peaceful integration or even the uneventful colonisation of vacant lands.

It's unlikely that there would have been much by way of a single, centralised authority in Britain in the wake of the receding Imperial authority. Imperial garrisons had been withdrawn from the North and North-West of Britannia some three decades before the 'official' Roman withdrawal in 410, leaving those areas essentially to fare for themselves. Archaeological evidence from The Wirral and from Eddisbury hillfort outside Chester suggest that there may have been piecemeal Irish settlement in those areas in the early Fifth Century, which would imply already a lack of effective defence in a former Roman stronghold. It's perhaps most likely that as Imperial authority receded, Sub-Roman Britain fell back upon those structures of pre-Roman authority which had largely been subsumed into rather than replaced by the Empire. There was already an established history of proto-English groups acting as garrison forces in Britain for some time before the collapse of Roman authority; for much of the 4th Century, the garrison of Vindobala among at least 5 other Roman fortresses, for example, had been comprised of Frisiavones recruited from Germanic peoples living between the Scheldt and the Rhine. An interesting example of probable integration here is in the kingdom of Kent.

Our earliest textual English source is the legal code of King Æthelberht of Kent, produced for him shortly after the arrival of the Augustinian mission to his kingdom in 597. In it, Æthelberht identifies as Kynges Cantwaras, king of the Cantwaras, a people identified by the Romans as inhabiting the region before their conquest. In instances of conquest or colonisation, Early English peoples often identified themselves explicitly, such as the Wæclingas who established their petty kingdom neighbouring a British community at St Alban's, so for Æthelberht to indentify specifically with a "pre-Roman" people suggests both that that was a meaningful surviving political and/or ethno-geographic distinction, and that Æthelberht's predecessors and their gesiths were able to integrate into rather than replace it.