You are specifying Anglo-Saxon. I have suggested in other threads a while back, the possibility of Britain retaining its Roman provincial structure as the Empire lost its cohesion, and some Brittanic-Latin dynasty deriving a claim from Roman administration and more or less controlling all of Great Britain south of Hadrian's Wall eventually replicating the OTL origins of the legend of Arthur. And this surviving Celto-Latin regime, perhaps allowing some Anglo-Saxons to settle but preventing them from taking over the whole island, might emerge, perhaps in lieu of Charlemagne, as the Roman Church's favored champion and be encouraged to claim the Imperial purple, but retain its distinct British features enough to insist on a new Imperial title, analogous to Caesar, but based on essentially "Arthur." The imperial rulers would thus be called "the" Arthur. I would imagine if they could keep Britain itself more or less secure, and use it as a resource and manpower base, they could claim generally prevailing if perhaps fluctuating control of western Gaul, extend it into the Low Countries, and more or less assert some power over Iberia, off and on, especially if they can do something to assist the Christian resistance to the Islamic invaders. With Church help and advocacy akin to how it attempted to aid Charlemagne, perhaps power could be projected into most of modern OTL France and thus connected to Italy, where the Popes would be keen to draw some fighting power to stabilize their position. Thus a Holy Arthurian Empire of the far west of Europe, variously adventuring into Scandinavia and with fluctuating fortunes in Iberia, might emerge.
The Saxons, if as OTL triumphant, would lack the sorts of preexisting ties to the Roman Church hierarchy a surviving Celtic-Latin Britain would, so it seems implausible the Church would turn to a power based so far west, or that the Heptarchy kings would much value a Roman framework for any high kingships someone among them might impose. It would take a while for them to become Christian at all in fact, just as OTL.
I think then if you want to have a British Caesar, you need to prevent the collapse of the Roman organization of Britain's provinces, draw them together under some Dux, and have this local dynasty energetically uphold and use the Roman structures and claim continuity with Rome itself, and thus have the power to contain the Saxon invasion. Sadly that means no England at all, or anyway it being limited to Anglia or some such; a Latin influenced Brittanic language or strongly British influenced Latin would form the base of the vernacular, I think the former far more likely, and Latin, perhaps evolving into a distinct form, the language of high governance. Perhaps Latin would be relinquished much as OTL to the Church and high scholarship and the business of the imperial realm would be centered, north of the Alps anyway, on a court refinement of the Celtic dialect would be the language of state. Which, if the Empire could maintain itself a few centuries or longer, would be disseminated among the ruling circles of all its territories, though insofar as the Lowlands and Scandinavia are drawn in, to them this would remain alien and local power there would do as much as they could in some Germanic dialect.
To survive, they would have to be strong enough to not be overrun too much by the "Vikings."