Ive always enjoyed the (admittedly eurocentric and over-simplistic) narrative of what we call western civilization originating in Greece, continuing through Rome, then into Britain, and finally into the United states. Almost all regions on the planet have been either controlled directly by rome or became colonies of areas that were once part of Rome. Combined with the widespread adoption of a US style constitution focused on democracy, the legacy of the west has touched nearly the entire globe.
It's obvious what the Romans took from the greeks, what the Americans took from the greeks/romans/British, but less obvious (to me at least) what the British took from the Romans. Just how influential were Roman ideas and technology on the development of britain? In what ways did the British "carry the torch" of the Romans? Naturally they deviated from the Romans in uncountable ways, but in what ways were they similar?
So, while it might be enjoyable for some to look back on their ethnic, national or cultural histories and gain pride from their accomplishments, it should first be mentioned that many of these histories were intentionally constructed, often quite recently, in order to further the aims of those invested in creating a nationalistic identity.
That being said, the narrative of Britain carrying the torch of Greco-Roman philosophical, technical or military knowledge is woefully inaccurate in almost every sense. The truth is that even during the height of the Empire Britannia was a backwater province, geographically removed from the beating heart of Mediterranean trade and cultural exchange and relatively sparsely populated. It had political importance as the home of up to 4 Legions - military forces provided significant political power in the Empire. But it lacked the wealth of the East, the cultural potency of Greece, the grain-producing capacity of Sicily or Africa and the connection to the all-important Mediterranean trade routes of Italy or Spain. The border with what the Romans called Caledonia was even relatively stable (not what Hadrians wall would have you believe, but there were far fewer/none of the incursions of the level that inundated the Rhine and Danube borders) and so little reason for Roman Emperors to pay it much extra attention.
Now I'm hardly an expert in the area, but from the research that has been done/is being done it is clear that Roman methods of agriculture, military/political organization and economic production pretty quickly dissolved in the wake of the crumbling of the Western Empire. This wouldn't have been a sudden shift, but rather a steady decline in the prevalence of Roman methods and technology. I once saw a post on this sub that made the analogy of how it would have seemed at the time. Instead of the garbage many coming every day as it had for years and decades, maybe he started coming every couple of days. Then as the years progressed it became a weekly schedule. Then maybe monthly. Then maybe more and more people started dealing with it themselves. People moved out of the urban areas into the agricultural hinterland in growing numbers. It would have been a gradual decline as opposed to sudden. But it happened nonetheless and was hastened by the instability caused by Anglo-Saxon incursions from the east. Anyway this is all to point out that by the 5th or 6th century the legacy of Roman Britain was in some ways mostly faded. The expanding Anglo-Saxons were not quick to adopt Roman titles, responsibilities or cultural norms such as some Germanic people had done further south. This is possibly a statement of their relative lack of Roman assimilation and the deterioration of Roman influence in the region starting before their arrival.
All that being said, the Romans were also capable of leaving impressive legacies without directly or indirectly changing modes of thought or culture. The Roman road system, for example, was hugely influential in deciding what areas were settled/brought under cultivation and governed trade networks for centuries. Important urban centres were often still inhabited to some extent, and as the population grew they often became important political/cultural centers yet again. And the legacy and growing power of Roman Christianity played a MASSIVE part in the development of political, literary and legal traditions.
Every culture and people build upon the legacies and traditions of those who have gone before. This is no different in the case of Britain as it emerged from the decline of the Roman Empire. But I would say, in direct answer to your central question, that the Anglo-Saxon kingdoms and their traditions had more of an influence on medieval Britain than Rome, and that the Norman conquest helped more to introduce Continental/Mediterranean/Roman language and influence than the Roman occupation had a millennia earlier.
Stuart Laycock (2008). Britannia: the Failed State. The History Press.
Francis Morris (2010). "North Sea and Channel Connectivity during the Late Iron Age and Roman Period (175/150 BC – AD 409)." British Archaeological Reports International Series. Oxford: Archaeopress.
Hills, C. (2003) Origins of the English. Duckworth, London.