A plausible theory is earlier migrations like you stated. The Celtic languages spread to Britain and Ireland from the European continent, and while it was probably largely the case of native inhabitants adopting Celtic speech and identity (akin to how people in Roman provinces like Gaul and Spain adopted Latin much later), I'm sure there was still substantial migration from the continent as well. The French and also Germans are largely descended from Celts/Gauls of the continent, which I've always associated the most with "Europe West" DNA.

Another source for you could be French ancestry via the Normans. Although the Normans were largely of Scandinavian Viking origin, they would've also been part-French by the time they invaded and conquered England as it was mostly Viking men who married French women to give rise to the French-speaking Catholic Normans. Many Normans did settle in England after the conquest in the period after 1066.