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It could well be something native to the Isles. The suggestion has been made before that pre-Roman Britain was split between quite different tribes, and the Silures were notably described as being [reasonably] dark-skinned and curly-haired like the Iberian peoples. In the sub-Roman period, new Kingdoms appear to have arisen quite quickly, suggesting a sense of 'belonging' to native tribes or culture had persisted over the centuries of Rome's rule. Perhaps it's the case that the rise in Mediterranean component is simply the result of the 'melding' of these different tribes over the last millennium-and-a-half, with modern Englishmen and Welshmen an amalgam of several very different Iron Age forebears.
That being said, migrations from the continent in the post-Hastings age cannot be ruled out. The Normans were certainly not just Vikings who had annexed Northern France, and certainly the Gallo-Romans of what is now Normandy were a major contributor to the ancestry of the people who rowed across the Channel in 1066. Your comment about the Normans being 'an elite who didn't leave much impact' is a common trope but I have doubts as to its accuracy.
First of all, the first wave numbered many thousands, amongst whom were not just Normans but others from throughout what is now France. Post-conquest, they weren't all rewarded with manorial landholdings and lordships. Normans occupied all levels of society, at least by the third or fourth generation after the conquest, and intermarriage between Englishmen and Normans is attested in chronicles of that period. It's not hard to imagine how this came to be; primogeniture can only benefit so many of a person's descendants, and so the occasional son or daughter of even the wealthier elite would have found themselves out of favour and drifting into the general population. That's before we even account for the later settlers, merchants and whatnot who would have come from France long after the initial conquest. We should also remember the Angevins' possessions, which stretched as far south as the Basque Country and with which England was in political union for many years (no doubt allowing for increased and easier trade between the regions).
I personally think this is a better explanation for the influence of Old French/Anglo-Norman on Middle English, rather than simply having a 'French' elite in England.
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