This is only reasonable with a rather limited scope I think. It certainly is the momentum of the moment, and it might be that the Anglo-Saxons will once again, and perhaps even again somewhat later, succumb to another Danish invasion at some nearby date, say the 1070s or '80s and perhaps again some century or so later--maybe. Politically speaking the ties will remain stronger than OTL, unless in the interim another closer to hand continental power manages to both preempt the Danes and drive off both native insurgency (or coopt the English powers that be in bulk) and alternative Danish attempts.
But even just culturally, and more so economically, England lies pretty far south to remain indefinitely "Scandinavian," and lies far closer to the Lowlands (southern Lowlands at that) and France/Britanny, with other economic interests tying the realm to Iberia as well.
Regardless of whether habit or even some ideological pro-northern chauvinism favor their thinking of themselves as having more in common with the realms of the northlands, and even remembering that after all some significant economic ties exist with the Baltic and north Atlantic along with those drawing them more southward, the overall balance of economic interests will veer farther and farther south as the general intensity of productive and mercantile activity rises.
Europe in the coming High Middle Ages remains more strongly centered on the Mediterranean than it would become in the Modern era; much of what is going on in terms of demographic increase and general elaboration of production and trade north of the Alps is happening in the HRE sphere, which is to say mainly Western Germany and the Low Countries, there again--mainly in the south in what OTL became Belgium. And France is also strong in this general sphere, and much of that, especially in tradable goods the English or whomever might lord over England would be most interested in trading for such as wine, is in the southwest of France. Meanwhile to obtain any trade goods, including spices and the like passed on by Mediterranean and overland traders (or by sea sailing around Iberia), they need to sell something, and so whether England is run by Anglo-Saxon overlords, or any foreigners from anywhere, they will become increasingly preoccupied with organizing England's own production to sell goods these mainlander foreign trade partners want--whether the traders are English or foreign is also a secondary factor.
Meanwhile by and large the way to bet is that the fundamental limits of agrarian productivity and thus population in Scandinavia also tend mostly to shrink the relative power projection potential of even the richest and strongest Nordic kingdom.
Conceivably if the entire Scandinavian constellation can be brought under one ruler who also takes control of Baltic trade very strongly, which might happen several ways, such a Scandinavian imperialist power might be able to leverage more gains along the southern Baltic coast and perhaps grab the northern Lowlands aka "Netherlands" as we know them today, or who knows maybe even round it out with gains of Flanders and perhaps on into northwest France and in such circumstances, gobble up England as well--at some point such a potentate would probably call himself an Emperor indeed and perhaps manage to strongarm the HRE mantle as well and thus acquire some claims all the way down into Italy and who knows how far southeast towards the Balkans and Constantinople?
But OTL argues strongly that such an extreme Scando-wank is just that, a pipe dream against the odds, with too many powers standing in the way of any single one getting such total control and with rivals with deep enough resources of their own to resolve to block it and break it. I can well concede that an intermediate outcome between such a wank and the OTL shrinkage of Denmark and Sweden could happen, say with a Denmark that holds at least some swathes of land north of the Danish straits to secure the whole passage, and perhaps gets a solid grip on a fair sized strip of north German and Lowland terrain for more demographics and some territorial depth which over centuries adopts a Danish identity and consolidated centralized governance. Or a loose but lasting Kalmar Union that either supplants or absorbs the Hansa--including the variant that what unites the disparate Scandinavian realms, perhaps with no personal union of their separate crowns even (these perhaps multiplied, into separate crowns for Trondheim, southern Norway, Denmark, Scania, Sweden, Finland and yet more for additional coastal kingdom-provinces, on the principle of divide and rule) is indeed the Hanseatic League gone more political, on paper a mere mercantile compact among legally separate realms but in fact with the League big shots in council and electing some Master with de facto monarchial powers to dictate what each of these diverse courts will in fact adopt as policy nominally and ostensibly on their own sovereign hooks.
No doubt the North could manage to win bigger than OTL on some premise or other, and to be sure they were no mean players throughout European history once the Viking age got going.
But there can be little doubt that even if some extreme Scando-Wank under Hansa masterminds or some fantastic concatenation of a core Scandinavian kingdom into a continent-spanning empire takes place, demographically, economically, culturally such a superstate would in fact be centered in mass well south of the Baltic to have such power, and meanwhile whether the northern parts are under some Baltic based regime or not, both they and the lands beyond to the south will remain collectively far outweighing the influences of Scandinavia proper.
England, and if we grant some manifest destiny for some realm or republic centered in the British Isles to eventually run the whole archipelago, whether on its own or as a satrapy of some larger system, the British Isles as a group, remain situated off the coast of western Europe and with both vulnerability to seaborne invasion from all quarters of it from Norway to Portugal, and vice versa the lure of positive trade relations (not to mention the temptations of coastal conquests for again either satraps of some continental power or an independent island based state) draws the attention of anyone lording it over any part of them to relations all up and down that sweep of European coast, and beyond if the British power or powers have taken the initiative of building up merchant and some degree of naval sea power, into the Mediterranean, not to mention the possibilities of colonizing the uninhabited Madeira and Azores and eventually, the western shores of the Atlantic in the New World.
Thus, even if it proves to be the case that there is a second and final Danish invasion and the establishment of such strong ties to Denmark that the realms never separate--which would I suppose bring in its train a consolidation of rule over everything lying in between, that is a wank of such a Dano-English realm to incorporate at least the demographic core of Scotland (possibly leaving some of the more wild parts of the outer Highlands on their own scanty hook, odds are they'd all be brought in eventually though) and on the European side, Flanders and northern Lowlands all the way to Jutland--still the sheer weight of demographics multiplied by greater per capita wealth as one goes south on either side of the North Sea will de facto make such a kingdom not so much Scandinavian as North Sea centric, and withal as a unified realm, with only a part of its attention north and eastward, that mainly focused on the east Baltic, either as another wing of satrapies to balance (but never I think overbalance!) the southwestern North Sea salient, or as a frontier where commerce crosses over to other sovereign turf and combined naval/overland threats preoccupy the great realm--which would often mean drawing heavily on the southwest wing to reinforce the northeast. But vice versa, the interests and attention eastward along the Baltic would be often preempted by strong interests and fears regarding the west wing, and the sheer number of subjects--most of them lowly commoners and perhaps serfs or worse maybe, but a substantial minority of them gentry and wealthy burghers with a major input into royal/imperial policy, whether that is constitutional on paper or not is not so important--on the Atlantic side of the Danish straits would be so high their whims will tend to at least match if not always trump the existential needs of the eastern wing's residents.
Under any sort of rule, whether under a government centered in London or some other English city, or under a strong central regime with center somewhere on the Continent, the interests of people living in England will increasingly preoccupy even a foreign-based government.
The only way I see that being minimized is if the foreign rule persistently scants the development potential of southern Great Britain, which would be against the personal interests of any Continental persons sent in to keep the necessary whip hand over the English subjects. If this is done, then I suppose the English tail might not wag the Danish or Hanseatic or French or Flemish/Imperial dog I suppose, but it happens at the expense of much potential wealth the grander power might well miss--therefore such a power on a course of grinding down England for the hell of it would be weaker than it could be against other rivals, and this might create opportunities for other powers, perhaps ones quite Lillliputian OTL, such as say a weakly united Irish conglomerate, or Scotland, or even a resurgent Wales, not to mention other Continental powers with designs of their own, to either foster an English secession or wholesale or piecemeal annex parts at least of England to themselves.
But that is unlikely in the extreme I think. Realistically, any foreign power that subdues England to its own rule will initially establish overlords who might indeed be very haughty versus their defeated English subjects just as William's Normans were, but over time, OTL suggests it might take centuries but those centuries will pass, these once foreign lords will become increasingly English--just as some of the most stubborn Irish resistance the Tudors and their successors OTL found in Ireland were nominally Norman barons who had "become more Irish than the Irish" in the interim. However astute a continental centered power might be at keeping up ties of interest to keep their overlord class in England secured to the larger realm, what this accomplishes, as the price of bringing English resources to the central pot of power enabling such a Channel-crossing regime to endure in the European cockpit, is to raise the importance of parochial English interests to the policy making body, be it an improbable unbroken succession of monarchial autocrats or some institutionalized more or less parliamentary council of the leading men of the realm. I'd also add that the odds are clearly in favor of the British satrapies splitting off at some point or other, in the course of a dynastic civil war perhaps, or debacle of the Continental power base battered by some strong mainland rival with the British holdings becoming a Taiwan-like rump, turning England once again loose as its own power base for better or worse.
In all of this, even if we come up with the most realistic Scando-wank we can balancing plausibility with serious aggrandizement versus OTL, and furthermore grant said power very astutely weaves together the interests of anyone who might rise to significant influence over general policy to always weigh the larger realm's interests in alongside their more specifically English centered ones, those notables and loyal commoners of this greater realm will remain engaged with projects that turn their attention to the Lowlands, and the French coast, and Iberia, and Mediterranean trade, and eventually transAtlantic and circum-Africa ventures into the Indian Ocean, and beyond to the Pacific--unless as noted England is so nerfed by perverse misgovernance it is dwarfed and stunted and perforce sits aside in rustic disability while someone else preempts these interests. Such a stunted England is as noted little asset to its haughty overlord whomever that might be and liable to be lost to them sooner or later.
Assuming then that England is run at least middling well for general profit and opportunity, its occupants, native and foreign alike, will have considerably more cosmopolitan interests that ballast any trend toward "Scandinavian" identity, offset only by either gross mismanagement stunting them, or by the Scandinavian identity itself also becoming grander and more cosmopolitan due to across the board managerial successes. In such a timeline, I suppose 21st century inhabitants of England might indeed consider themselves somewhat "Scandinavian," but only because that identity largely takes the place of "modern European" broadly, and we will find that ATL Danish or Swedish or Norwegian is itself Frenchified, and generally Europeanized with borrowings from German, Spanish, Italian, and of course Latin, alongside perhaps some wide currency of Slavic and Baltic languages and bits of cultural practice scattered all over northern Europe and filtering south too.
Whether this happens in the Baltic zone or not, in England at any rate, a lot of this sort of borrowing will be going on no matter how Germanic England's overlords, whether Anglo-Saxon in unbroken succession or involving a bunch of Danish and Flemish adventurers as well, have remained in the ensuing thousand years. Half a thousand years before 1066 after all the earliest Anglo-Saxons of the Heptarchy were already showing a distinct propensity to borrow words which suited them quite a lot even with no swords to their necks. Surely we would not see any of the OTL direct borrowings from specifically Norman French (unless the Normans make a comeback and try again and succeed this time--but while the OP is very vague and we can't categorically rule that out, it runs pretty counter to the implicit basic OP question so let's dismiss that and figure the Norman wheel is well spoked--this doesn't rule out a later French invasion of course) and might given some other foreign conquest see Danish or Flemish (or Parisan French, or Angevian, or Gascon or conceivably Portuguese or Castilian taking their place) while if it remains Anglo-Saxon all the way, pretty much, older words would endure that were transmuted or displaced completely OTL.
But eclectically it is the English propensity to steal whatever words strike their fancy and generally not even bother to file off their foreign serial numbers except by a process of centuries of accustomed use making these plainly foreign borrowings (more often than not, not even Germanic and perhaps not even Indo-European, especially once the English under whatever flag and allegiance are roaming around the globe) make them just as commonplace as the most Saxon pedigreed words in the OTL lexicon. Evidence of this sort of cheerful cosmopolitanism abounds already in the Saxon period and I don't see how it would be otherwise in any ATL except for the possibility of a deliberate revival of an English driven to near extinction much as Hebrew and Gaelic have been revived OTL--and I suspect such near extinction is very unlikely to happen in the first place.
England could be very different if William's Conquest is averted, but I don't see any warrant to suppose it remains frozen in North Germanic amber, not even if North Germans such as the Danes conquer it more thoroughly and lastingly than even the Normans managed. It is not in their ruling interests nor those of the English subjects, be they serfs or satraps, to so divert and thus probably stunt the straightforward benefits of such a conquest in the first place.
England remains poised to sample the entire array of what the Atlantic coastal European platter, and eventually beyond to the entire coastal world, offers, whether we call it a "buffet" or a "smorgasbord."
