I appreciate that this is a VERY broad and general question, and every one of your answers will be accompanied by an unavoidable "It was more complicated than that" but I think you can grasp my general meaning.
From 1066 onwards, we refer to the ruling elite of England as the Normans, as that's who they were yet by at least 1200, we stop this and simply refer to them as "The English" in the same manner as their subjects. Thus, the question becomes;
When did this change occur, how did it happen and why?
Why do we stop using the term Norman to describe the ruling elite of England, despite the fact that they are the ethnic descendants of the Norman Conquerors, spoke French as a primary language and even until 1689, the idea of the monarchy being the "Norman Yoke" was a common idea in England? Was this something they recognised themselves or do we wrongly apply the name "Norman" or English" to those in the same way as we - arguably - do to the "Byzantines?"
When do the similarities with the original Norman conquerors end and what is the point where we can refer to a collective English people, rather than an alien elite and a subdued populace?
The identity change of the Normans and when they began being referred to as 'English', interchangeably with their indigenous English-speaking subjects, was gradual and by the 1300s and conclusion of the Hundred Years' War, the Norman identity in England was replaced by the English one.
The loss of the Duchy of Normandy in 1204 by King John to the armies of Philip II of France, and subsequent collapse of the Angevin Empire in 1214, was the first event to be particularly damaging on the Anglo-Norman identity in England. In 1259 England gave up all claim on Normandy to France. With their namesake 'homeland' lost, the Anglo-Normans were left with just England and naturally they assumed a more English identity and England centricity. The Normans were heavily invested in England, since it was their largest most-prized possession and their only Kingdom.
The French crown meanwhile assumed control over Normandy and the ducal title was transferred to the French King and passed down to close members of the royal family. The French Kings wanted to assert power over Normandy, and they did this by constructing large fortresses such as the Château de Rouen but in their mixed-approach the French King also granted the Charte aux Normands (Norman Charter) to win over the loyalty of the Norman population from supporting Norman-ruled England.
During the Hundred Years' War, Normandy was re-taken temporarily starting from 1345 and changed hands several times until ultimately being conquered by France once again in 1450. Throughout the course of the war the Anglo-Normans had taken steps to distance themselves from the now hated French by establishing and promoting English as the official language in 1362. Great English literary works such as the The Canterbury Tales by Geoffrey Chaucer were authored during the period and this led to a popularization of English in mainstream literature. In 1399 Henry IV was the first King to take his oath in English and his successor Henry V was the first Anglo-Norman King to write in English.
With regards to language use, Anglo-Norman continued to be spoken past the 14th century by the ruling elite and in commerce, however the identity of the Anglo-Normans had permanently changed. Most of the Norman ruling elite, at least by 1150-1200, were able to speak the indigenous English language along with their native Anglo-Norman language. By the 1400s certainly all Normans in England spoke more English than Anglo-Norman. But language wasn't the cause for identity change, it was politics and cultural assimilation.