How many grains of sand makes a pile a pile?

Since this wasn't a one-day change, and there are many ways to define the issue, I'll be using the following comment of yours as the reference point:

Perhaps when the royal court spoke English rather than French

The problem with this is that in 1065 they spoke not French but a variety of Old French we call Norman. In 1067 they were already removed from the rest of Old French so starting at that point, their "French" started to change, just as it changed the Anglo-Saxon language that became modern English. Furthermore, in 1204 any holdings they had in mainland Europe were lost with the fall of Normandy to King Philip II, and the Hundred Years War (1337-1453) pretty much put the final nail in the coffin for French's status in England.

By the next century English was doing much better in terms of PR with people like Chaucer taking the lead in creating English language literature (as opposed to the language primarily being used for speech and not writing).

However for the nobles, we're back to the issue of gradualness. While we can point to a few dates and say "Ah ha! This is when things changed", we actually have plenty of counterexamples. For example the Provisions of Oxford, an early legal document from the mid-13th century — 1258 specifically — was produced in Middle English, a significant thing for the time. It marked the reintroduction of English in that role. But it was simultaneously put out in French and Latin, so it wasn't like there was a clean break.

Really what you have is, rather than a changeover from one language to another, you have a gradual shift combined with a convergence of the two languages to become what we now call Middle English. That's not to say Middle English is actually a creole. It's not. But at the same time that Early Middle English was getting Frenchifyied, the speech of the Normans was Anglicising.

tl;dr: There's not actually one point. The shift was gradual and multi-dimensional. It's grains of sand.

Well, the English were English before the Norman Conquest- since 927, in fact, when Aethelstan united the English people. So you need to consider- what exactly is "English"?

You could break it down into different aspects- language, literature, material culture, laws, and territories. Language is the obvious one, but remember that French was a collection of dialects (the northern langues d'oïl), as was English- so that even in the 1380s, Chaucer could write that "ther is so gret diversite/ In English and in writyng of oure tonge" (Troilus and Criseyde). The legal system and church dealt in Latin, but also in Anglo-Norman. French was the language of the court- you could perhaps date the end of Anglo-Norman to the reign of Henry IV, the first to take the oath in English. And of course the languages took on elements of each other- so that loan words either replaced or co-existed with existing Middle English terms.

In terms of material culture- well, churches and castles in particular were directly influenced from the continent, in some cases (e.g. the White Tower of the Tower of London) importing not only architects and styles but materials (Caen stone in particular). Both Romanesque (11th/12th C) and Gothic architecture (12th C onwards) had their roots in the French territories. But in the form of Perpendicular/English Gothic, you could say that an English identity was formed by the mid-12th century, where the master-mason who completed Canterbury Cathedral known as "William the Englishman" in comparison to the Norman masons who preceded him. English tracery focused on arches; continental tracery focused on rose windows (the Rayonnant Style). But that's by no means a hard and fast rule.

Likewise, holding land in France is awkward- the Hundred Years War was responsible in part for the creation of a distinct English identity, but it was fought over control of French territories. Mark Ormrod (The Domestic Response to the Hundred Years War) described it as ‘a form of national crusade', although that has its own issues- were the antagonists aware that they were fighting for a distinct nation? Or was it, as scholars such as Duncan Hardy suggest, more of a dynastical claim with a side-effect of nation-building? And it's not as if English possessions in France were exclusively or even predominately from the original Norman inheritances- the possessions in Gascony were brought in not from the Anglo-Norman inheritance but through Eleanor of Aquitaine in 1151. Calais remained in English hands until 1558- long after England could be seen as the established identity of the ruling classes. Plus, it's not a steady change- was Richard II, born and raised in France, more French than his grandfather Edward III? Did he see himself that way?

So it's less a point than a series of points- formed across a wide range of different aspects, sometimes shifting back and forth. You can point to where it started- and an end point where you'd strain to pick out an Anglo-Norman identity. But where the tipping point is, is beyond the scope of a Reddit comment.

That depends on your definition of "English". Defining who is and isn't a particular nationality has never been simple. At what point does a French person who moves to London today, become English (or at least, British)? Now, maybe today that answer would be "when they become a UK citizen", but such a thing didn't really exist in the middle ages. So, what definition of "becoming English" would you consider? That they spoke English? That they considered themselves to be English and not Normans? Or just when they had lands in England and no longer had lands in France?

You could consider the end of Norman French rule to be when Henry the first died. Henry was the last true Norman king of England after his death the Anarchy began which was a 19 year long succession war between his daughter Matilda and his nephew Stephen. Matilda was Married to Geoffrey Plantagenet who was the count of Anjou. Matilda eventually won and her son with Geoffrey Henry the second took the throne. Even though the kings of England would still speak french for several hundred more years and the still controlled Normandy. They where not Norman but Plantagenet.
P.S most of my information comes from the book "The Plantagenet's" by Dan Jones, great read.