An excellent paper from the English Historical Review asks, "Why did the Anglo-Saxons not become British?" (H/t: Medievalists.) The question may be of interest to us today, not merely for historical reasons, but because of our own mass immigration from very different cultures than our own: different in language, education, poverty, and so forth. What prevented the Anglo-Saxons from fading into the British culture?
The 19th century idea, which the paper considers and rejects, was that the Anglo-Saxons had simply destroyed the native population or driven it out -- rather like early Americans did with the Cherokee. We did not become Cherokees because we cleared the land of them and their civilization, and thus there was little chance that we would intermarry or become subsumed. (I'm told that, in Latin America, genocide against the native population is euphemistically called "the North American solution" to the problems their cultures still today face integrating Spanish civilization and the native ones.) The 19th century thinkers found this view plausible because of the racial theory of the day. Unlike other peoples, those descended from good Anglo-Saxon blood were of a moral character fit for self-government; and since much of moral character was thought to be carried in the blood, obviously the Anglo-Saxon bloodline must be substantially different from, say, the Welsh.
That turns out to disagree with both the historical record and the genetic one. What happened instead was that the Britons intermarried with their conquerors. Rather than the Anglo-Saxons becoming British, the British became Anglo-Saxons in those areas where the new culture could command. In regions further flung, the native culture continued to dominate in spite of intermarriage.
Importantly, then, the issue has to do with language. So why did the Anglo-Saxons not become Frenchmen after the Norman conquest, when French became the official language of court and law?
In an important sense, they did: they became "frank" and free men who were fit for the "franchise." Their old thanes became "franklins." Amusingly, the very concept of being a people whose moral character fitted them for self-government that was fielded by the 19th century scholar with pride in his Anglo-Saxon roots was a French import. It was because they were "frank" that the English were fit to be free men.
(And that of course is true, in a way: the virtues of honor and forthrightness that the Franks saw in themselves are indeed a necessary condition for sustainable self-governance. What is not true is that they are especially French virtues, or especially Anglo-Saxon ones.)
The French language didn't come to dominate in England the way that English did, however. There simply were too few of the Anglo-Normans, and they spread themselves thinner yet: they went to Scotland and became Anglo-Norman-Scots, like Robert the Bruce, whom today we think of as the very model of a Scot. They went to Ireland and became Anglo-Irish, and while today the Normans are referred to by the Irish as "the Old English," even by the day of Elizabeth I they were more Irish than the Irish themselves. They were also the locus of Irish resistance to English rule, as they were the locus of Scottish resistance to English rule. I suspect Tolkien's concept of Black Numenorians is rooted here, in spite of his own stated preference for "Anglo-Saxon" characteristics more like the Eorlings and the Hobbits than like his Numenorians. The Normans like the Numenorians went everywhere the sea would take them, conquered and intermarried, and came to be the ruling element of different and disparate peoples. In the West, they were the leaders of the defense against Sauron; in the East and Harad, they were commanders of his legions.
The lessons for today are... well, again, obvious, aren't they?