I posted here about how quickly "English" French became a surprisingly different language from the French being spoken in France. This was the result of William I's nobility being an amalgamation of people from various different corners of France, all with their own dialects and languages, who also partially adopted the linguistic structures of the Old English language of their structures. Within a generation, Anglo-Normans were speaking a quite unique language which would eventually become Middle English.

William I actually took great pains to paint himself as the legitimate and rightful heir to the throne of England rather than a foreign invader. Indeed, he even originally planned to leave England and Normandy as separate entities to his sons Robert and William. The Normans themselves integrated themselves into English society quite thoroughly. Their legal reforms, especially the forestry laws, did a very effective job of alienating large swathes of the actual English population, but the Normans established themselves as part of nobility to a great extent.

In particular, they married into many of the established English noble families in an attempt to present a facade of continuity. Many of the prominent English writers of the late eleventh and twelfth centuries were of mixed Anglo-Norman heritage. Men such as William of Malmesbury, Orderic Vitalis and Gerald of Wales typified the new Anglo-Norman nobility, typically with Norman or French fathers and English or Welsh mothers, and who took great pride in their mixed heritage and new English identity. In their writings, within a generation of the Conquest, there's already a definite sense of English identity, both ethnically and linguistically distinct from France.

Orderic and William in particular aspire to an English identity in their histories. Both claim to write in the tradition of Bede, and use his Historia Ecclesiastica as a model. William of Malmesbury writes his Historia Regum Anglorum and draws heavily on the Anglo-Saxon Chronicle and Asser's Vita Ælfredi among other earlier works as his sources.

Another example is Richard of Hexham's account of the Battle of the Standard in 1138, when a scratched together Anglo-Norman force defended against a Scottish invasion of Northumbria during The Anarchy. Richard of Hexham's account in particular emphasises how the English army was made up of Anglo-Norman knights and Anglo-Saxon infantry. The Scottish army was in fact also comprised of Scots and Normans, since David I of Scotland in particular had been aggressively "Normanising" Scotland. I posted a bit about that here. Interestingly, David I was the great-grandson of Anglo-Saxon King Edmund Ironside, and was married to the great-niece of William I. His sister Matilda was the wife of William's third son, who would become Henry I, and the Battle of the Standard was fought to try and support his neice Matilda's claim to the English throne. Eventually, her son would be crowned Henry II, while David I's line would form the House of Stuart which took the throne of England in the seventeenth century.

Some reading:

Britnell and Thatcher, Progress and Problems in Medieval England

Davies, R.R., Domination and Conquest

Gillingham, The English in the Twelfth Century