Any chance Harold survives at Hastings, William is killed instead, and the Saxons decimate the invaders in detail, then take ships of their own and invade Normandy (and allied territories, I believe Flanders was involved too), take it and hold it?
That would be in defiance of the King of France's claims of course, but feudally speaking as a fief of France, that king had responsibility for William's actions. If the English can seize Normandy, and defy the French attempts to get it back, they can claim the territory as conquered. If successive English kings value it enough, as a foothold on the Continent, to beat off repeated attempts on it, I can imagine a reversal of OTL--English lords over a Norman-French peasant substrate. With the easiest invasion ports to England in their hands, England itself is relatively secured, able as OTL to mobilize what wealth the realm can produce toward maintaining the best navy around and with resources, augmented by exploiting Normandy and Flanders, to support an army largely based in Normandy. The monarchy is split in attention between managing England as "home" and Continental wrangling, which a sufficiently wise succession might indulge judiciously, calculating their strength beyond what it takes to maintain their footholds so as not to risk losing that. The fact of the face of the kingdom, politically and militarily speaking, being turned toward France and Continental intrigues "Frenchifies" English high society and paves the way for a reverse modern English fusing Norman French with island English, giving a cachet for things French to the island notables, so gradually English and the evolving Anglo-Norman vernacular tend to converge at least superficially. Can something parallel to Middle and modern English evolve from a pidgin? I know of other languages that have.
The upshot I suppose would be for southeast England preoccupied with trans-Channel business and Normandy itself to evolve the kind of English we know best today, and for England in general to be more "Up North" and quasi-Scottish--more Sir Gawain and the Green Knight, less Malory.
Geopolitically I suppose this is a dangerously unstable arrangement, dependent on the English kings never overplaying their hand. Could quasi-parliamentary institutions evolve constraining the monarchs to either maintain a certain level of professionalism, reliably, perhaps through evolution of semi-elective succession--the princes of the royal family know that for the heir apparent to actually succeed to the throne they have to meet standards, some body of notables customarily deemed competent to judge rules on it, so the designated heir has a strong motivation to comply, to demonstrate the necessary balance of political competence, with the court also keeping an eye on ambitious younger sons, finding honorable places (Lord of the Admiralty and the like) for decently competent ones and making sure both that they stand by ready to become king should it be necessary but do not plot to plunge the realm into civil war to try to usurp it either?
This is a tall tall order of course. But it might not be inconceivable, given England's island base where the rival kingdoms to worry about are limited to Wales and Scotland; perhaps there is early union with Scotland, and Wales I suppose must succumb, perhaps on somewhat negotiated terms involving dynastic union on a theoretically equal basis. That leaves Ireland, which is surely not going to unite and form a very dangerous rival. If some continental enemy were to gain sway over Ireland, that would be bad, but communications are by sea, which an evolving English navy might interdict, and eventually policy might put Continental distractions on hold to turn toward securing Ireland to prevent such annoyances, perhaps on rather loose terms akin to Finlandization--some cadet dynasty maintained in loose overlordship at Tara, mainly focused on preventing Continental rivals from carving off their own footholds.
With the British Isles thus secured, as a permanent deep pockets fiscal and demographic resource, perhaps an ongoing hold on a well defined English Conquest set of territories can be maintained despite the high cost of being vulnerable to strong land armies. Expansion is a temptation and a danger, but perhaps it can be done in a methodical and irrevocable fashion, say by dynastic union with Anjou securing the whole French coast and north from Flanders, the Low Countries to say Jutland. This would be a crushing stunting of France of course, and might lead to drawing the English farther east to seize Paris as well as the eastern, Mediterranean parts of Languedoc from an Angevin-Gascon base. Such expansion if sustainable would leave Normandy near the heart of the whole system. I don't think the English would move the capital from London though, London is just as central and convenient as a natural center of their English heartland and naval/mercantile base.
In addition to sheer overexpansion and vulnerability to heavy attack from a rival continental power (with France crushed or perhaps even absorbed, this means basically some version of the Holy Roman Empire I suppose) there would be the danger of sheer centrifugal secession. A strong local lord, discontented merchants and landowners bypassed by policy in London, an ambitious younger son (or daughter!) of the royal family, deep social fissures such as led to the Reformation OTL--the low integration of late medieval and very early modern society versus the levels we assume today, might create too many centrifugal tendencies to split off the various parts again, unity be damned.
Certainly OTL the Normans maintained a trans-Channel regime a very long time, they and their successors. But only for 400 years or so. Can Anglo-Normandy be maintained as a bastion by an English system taught prudence in avoiding the siren song of empire, just doggedly holding on to one or two large bits and stoically enduring the imperfection of their security with Scotland and Ireland as well as a strong France and HRE all facing this realm on all sides indefinitely?
At some point, the union between England and Normandy can rupture and Normandy still remain "English-speaking," if it is possible to get to that point in the first place. But I wouldn't want that split to happen until say 1600 or perhaps later to keep the linguistic shift set. Even at that, the reversed nature of the mixture of English and Norman French would be such that countryside villages will remain basically French speaking with a lot of English overlay.
A language that is fundamentally Romance in basic grammar and commonest utility words, the way English of OTL remains Germanic despite the fact that most words in the dictionary have quite different origins, is not "English." The way for it to be English would have to involve the overall pattern paralleling that of Norman overlords pressing down on English peasantry. So I actually am thinking of two hybrid languages here--a French-overlaid, basically Saxon Middle English which as noted happens because English lords are culturally captivated by French culture; this can take root in Normandy itself only to the degree that English commoners migrate into Normandy to seep in demographically, perhaps in a pattern similar to the founding of German towns scattered all through southeast and farther eastern Europe in a non-German countryside--the English being privileged and politically reliable tend to become the burghers and urban artisans, albeit with a lot of countryside Normans assimilating to them. And there would be a separate (though superficially fused seeming perhaps) Normanish or Normande, a reverse deal where the basic common words and grammar remain stubbornly Norman-French, but a rich overlay of English words and some grammatical mutations give it a superficial resemblance to Middle English. This is why I think it is important the English actually remain in control, Normandy firmly and perpetually under rule from London; this privileges the actual English speakers--a minority, but a politically and socially dominant and important minority. If at any point union with England were shrugged off, as late as say some ATL expy of the aftermath of the Great War, the stubborn persistence of the peasant back country asserting Normande ascendency might purge the actual English speakers. Perhaps some political development can create a national identity intertwining both so that both mirror image hybrid tongues persist, but frankly I'd think one or the other would tend to dominate in time. That might be the English dialect I suppose, with only the most rustic in the more peripheral villages failing to switch over to the Germanic base even as the newly separate and self-identified Norman land celebrates its separation from English rule by superficially adopting all sorts of Normanisms formerly deplored by English-ruled high society.
Now perhaps instead the fusion and eventual dominance of the Germanic version with its commonalities with the English dialects on the home island itself can happen early enough that even with earlier fission, Normandy remains an English speaking nation, albeit increasingly divergent and subject to going more French in vocabulary. Maybe just a few centuries can do it? My main reason to doubt that is that the Normans hardly got the English speaking French as such OTL, except as a second language, so it is tricky to see the mirror image situation acting deeper or faster in Normandy unless either the situation is less symmetrical, or the time span is much longer.
I think it would actually help Englishizing Normandy if there is never any policy to extirpate the French-based country dialect, if in fact it is more than tolerated, but even facilitated. Certainly it would be useful to an English kingdom with a Continental foothold to have a decent sized base of subjects who are reasonably loyal yet speak natively a form of French; this will help with communications on various battlefields and in recent conquests at French expense. If persistent French is respected in Normandy, while meanwhile English burghers and city artisans form a solid demographic core for English as such in the province, perhaps voluntary self-assimilation will over time accomplish what a thousand years of brutal forced assimilation never would.