The period between the Romans leaving and the re-establishment of 'order' on the island is ,for obvious reasons, a cloudy one. The bulk of our evidence is not literary, but archaeological and thus requires a great deal of interpretation not guided by written records. For example, we do know that some trade continued between Britain and Europe, as evidenced by grave goods and found hordes. The best 'contemporary' writer we have is Gildas, and he writes of Saxons being used as mercenaries or foederati on the same model as the rest of the Empire, and then those mercenaries trying to take over from their employers. He is not really a contemporary to the events he is chronicling though, so it's pretty much a wash as to exactly how accurate it was.

However, if we take other parts of the Empire after the legions withdrew as inspiration, we might have some ideas as to how it might have worked in post-Roman Britain. In most of the rest of the Empire, villas that once served as the palatial estates of the rich were now transformed into places of refuge, with their small walls serving as bulwarks against raiding parties. A somewhat 'Roman' way of life likely continued, with some education and cultural practices being retained. Whether the Saxons were invaders or took over from existing Roman elites in a more natural settlement, Romans/Romanized Britons themselves would likely have been retained in administrative roles such as they were, i.e. things requiring literacy. Your 'leader' in a secular sense would, assuming you were a Briton, likely have been your local landowner who owned the villa and whose fields you worked and whose protection you needed. If you were the landlord, it's more difficult to say. Would you have been basically independent, or was there some sort of kingdom structure erected? We don't really know. Gildas suggests there were kings, but the actual men he describe seem more allegorical than factual. Because the Roman Empire itself placed a great deal of importance of local governance by the wealthy in the provinces for non-military matters, it is likely that this system simply carried over into post-Roman Briton (for a time), albeit without the force of arms of Rome to protect it.

Michael Jones's 'The End of Roman Britain' is a good summary text of this view, although he is perhaps overly pessimistic about the ability of Britons to govern themselves. Another supporter of this is from one of the more prominent authors in this field, Stuart Laycock. He suggests in a couple of his books that Briton itself was riven by tribal strife before, during, and after the Roman prescense, and the governmental situation in the 5th century was chaotic infighting which the Saxons took advantage of. There are also quite a few other historians who, I think capitalizing on the relatively recent uptick in Arthurian interest, attempt to relate that mythical king with possible real people and situations. Your mileage may vary on those works.