As for the fate of political elites (large-scale English (Saxon) landowners), you can check the previous posts by /u/BRIStoneman that I linked in: How was William the Conquerer able to effectively supplant the Anglo-Saxon nobility with Norman nobility? What happened to the old nobility? /u/welsh_cthulhu's answer in the linked thread might also be interesting to you, though the primary focus on the post was the Welsh, not the Anglo-Saxons.
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As for the fate of Anglo-Saxon clerical elites, I suppose that careers of Archbishop Stigand (d. 1072) and Thurgot of Durham (later Bishop of St. Andrews) (d. 1115) are especially telling ones.
First of all, it should be underlined that William drastically changed his policy on the old Anglo-Saxon clerical elites around 1070 (during the Northern Revolts), not right after what we call Norman Conquest in 1066 (that can also be applicable to his relationship with the former Anglo-Saxon local political elites).
As I summarized before in How true is it that England was briefly Orthodox before the Norman Invasion?, Archbishop Stigand of Canterbury had certainly very powerful as well as controversial figure in the last decade (exactly speaking, since 1052) of pre-Conquest England. William the Conqueror, however, chose to ally with him at first in the first year of his reign in English, with his colleague, (Anglo-Saxon) bishop of York, Aldred (Loyn 2000: 60f.). Exact contemporary sources like Anglo-Saxon Chronicles tell us that Anglo-Saxon Bishop Aldred crowned William as a new king of England in the Christmas of 1066, possibly assisted by Stigand. After the coronation, Stigand also accompanied William to Normandy in 1067, together with some other Anglo-Saxon secular magnates like Earl Waltheof.
About the same time as the Northern Revolt (1068-1070), William chose to ally rather with Pope Alexander II in Rome who was especially harsh against Stigand due to his alleged un-canonical status - Stigand had got pallium from Alexander's uncanonical predecessor, Pope Benedict X. Papal Legates landed in England, and convened Whitsun Church Council in Winchester in 1070 where five remaining Anglo-Saxon bishop were severely checked and Stigand was finally deposed (famous Lanfranc of Bec, ex-tutor of Pope Alexander II took his place as Archbishop of Canterbury). It was rather a series of church councils and related actions in early 1070s that decided the shape of Anglo-Norman church hierarchy in England under William and his successor. In this context, it is worth noting that even the almost contemporary text on the Conquest, such as William of Poitiers, write the events retrospectively after this alliance of Pope Alexander and William, so, as I briefly mentioned before in: Why did Pope Alexander II support William's claim to the English Throne?, we should be careful of trusting too much on such allegedly good relationship between the Normans and the Papacy, and their description of 'villain' Saxon prelates in these texts. Sometimes Norman Conquest is summarized as "Europeanization/ Standardization" of Anglo-Saxon England even by some historians, but it was not simply so, at least in the wake of the conquest in 1066.
Then, it's time to move to the early career of Turgot of Durham. Young Anglo-Saxon clergy Turgot, native of northern England, was caught by the Normans during the Northern Revolt, but somehow he managed to escape from the custody and took a ship to Norway around 1070. Symeon of Durham narrates that Turgot was warmly received there in the royal court of Norway under King Olaf Kyrre (r. 1067-93), son of famous Harald hardrada, and Norwegian nobles.
He had served in Norway as a kind of court chaplain for about 4-5 years, but he returned to England and took a monastic vow. As a prior of the monastery of Durham (1087-), he became the secondhand of bishop of Durham, and got gradually familiar also with Scottish royal family (He was known to have written the biography of Queen Margaret, wife of King Malcolm III of the Scots). In 1107, he was elected further as a bishop of St. Andrews in Scotland under the auspice of King Alexander I of Scotland as a culmination of his involvement with the people on the fringe of the Latin Christendom/the British Isles.
Possible contribution of the Anglo-Saxon clergy in the process of Christianization in 11th century Scandinavia was the theme that attracted recent attention from researchers, and Abrams points out the possibility that it was not the Anglo-Norman, but the rather "old" Anglo-Saxon clergy, like fleeing Turgot in his early age, still played an important role in it across Scandinavia in late 11th century (Cf. Abrams 1995).
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