This question has been asked before but never got any responses from an expert on the period. My own knowledge ends around ~700, but I think it will be still be relevant, since in the seventh century anyone who was anyone knew about the Romans - I see no reason why this would change later, especially as contact with the Continent expanded further in the mean-time as the Carolingians consolidated their power. At the very least, I don't think Egbert of Wessex would be the only person interested in their island's Roman past, as he is portrayed on the show. My comment here therefore does not answer strictly your question, but I hope it will add an additional insight to any other answers that you may receive here.

The first Anglo-Saxon historian of note, the Venerable Bede, lived and wrote in the early eighth century and purely from flipping through my copy of his Ecclesiastical History, it is clear that he knew about the wider world - he for instances recorded the names and reigns of Roman emperors down to Leo III (717-741). This is not unusual, since from letters and other records, we know that Anglo-Saxons regularly travelled to Rome (and sometimes beyond) to gather manuscripts and relics to take back to England, including of course historical accounts of the empire. There were some pretty well-travelled people from this time, such as Arculf, who went all the way to Jerusalem in the late seventh century, as well as Boniface, who was a missionary working amongst German pagans and who had regular contact with the pope in Rome in the eighth century. This is quite important, since the city of Rome was still in theory within the Roman Empire ruled from Constantinople, so visiting the city was in effect visiting the Roman Empire, whilst visiting Jerusalem involved passing through Roman lands and lands under Arab rule that had only recently been taken from the empire.

Bede's account, aside from his up-to-date knowledge of continental affairs, is also interesting because he stated outright that his Ecclesiastical History would be read by his king, Ceolwulf of Northumbria. If the king ended up doing so, he would have read about Julius Caesar's ill-fated invasion of Britain, Constantine I's flirtation with Arianism, as well as the danger posed by Attila the Hun - basically all the highlights from Roman history. Bede had at his disposal the chronicles and histories dealing with both recent and ancient history, so naturally he summarised them in his work. I don't know what happened after Bede, but his work was always extraordinarily popular and copied often, so presumably many educated individuals would have had a basic understanding of Roman history just from him. Even if this is incorrect, I doubt that Bede's sources were lost or that Anglo-Saxon travellers to the Continent failed to pick up some basic Roman history/manuscripts on the way, especially if they actually journeyed into the empire itself.

There are also other more interesting connections from the seventh century, the influence of which can at least be felt in the early eighth century as well. The best example is Theodore of Tarsus, a monk from Cilicia deep within the Roman Empire, who became the archbishop of Canterbury in 668. His deputy, Hadrian, was in fact a North African monk who had once served as an ambassador to Francia on behalf of emperor Constans II. These two men brought eastern learning to England, such as the theology taught in the city of Antioch and hagiographies of eastern saints (which included the cult of Anastasius the Persian, a convert from Zoroastrianism who was martyred by the Persians), as well as the popular collection of ascetic sayings/anecdotes collected by John Moschus, a very important eastern monk who compiled it c.630. Theodore and Hadrian founded the famous school of Canterbury, transforming England briefly into the 'light of the West', drawing students from both Ireland and Francia with a curriculum firmly based on the classical past and the most up-to-date contemporary knowledge (such as the work of Isidore of Seville from Visigothic Spain). The thing that impresses me the most is that these students were coming to England to learn Greek, which just goes to show how interconnected Christendom remained in this period, even though there is still a tendency for some people to see it as the 'Dark Ages'. Until the school of Canterbury gradually petered out sometime in the early eighth century, England was actually a hub of Roman learning!

This connection is even more impressive when we realise that Theodore had also been involved in the monothelete dispute (a doctrinal debate within the Roman Empire in the mid-seventh century) and was personally affected by the Arab conquests, as he had fled his native Cilicia to the city of Rome because of the danger posed; as a result, the Anglo-Saxon Church in the seventh century was very much aware of issues within the Roman Empire and its many councils serve as evidence of how keen they were to ensure that they also had a say in them, most prominently by backing the papal doctrinal position in 680 against the pro-monothelete elements then still active in Constantinople. Indeed, Theodore was a very important player in the rebellious Lateran Synod of 649 against the emperor Constans II and was still considered an important authority on countering monotheletism by 680 when the pope needed help to finally crush the heresy. Hadrian's career was even more important, as aside from being an ex-imperial ambassador, I think there are reasons to think that he had turned against Constans II by 668 and that his disillusionment represented a larger wave of discontent within the empire, which culminated in the emperor's assassination (with a bucket) the same year. As such, Theodore and Hadrian were deeply involved in and aware of the crisis the empire faced. As the Arab conquests marched relentlessly on, two individuals caught up in these extraordinary events ended up in England, how amazing is that?

This was also the age of Wilfrid, an influential Northumbrian monk-bishop who travelled to the city of Rome three times. Wilfrid was a firm Romanophile and he brought Roman liturgy, relics and craftsmen back to England. He had also been affected by the furore over monotheletism in Rome and I think it played a major role in his later conflict with Theodore; both were pro-papal figures, but their experience with Rome taught them different lessons about how to promote Roman orthodoxy in England. These two men dominated Anglo-Saxon ecclesiastical politics in this period, so it is rather important that they had both been in Rome and were aware of the twists and turns of imperial politics. The battle between Wilfridians and the moderates would rage on into the eighth century, so again, it's a clear case of people with knowledge of Rome continuing to be influential throughout this period.

These individuals in the seventh century and their impact are literally my favourite things to research, so please let me know if there are any questions and I'll do my best to answer them! :)