The longer a particular writing system is established the greater the need to change it will have to be. What benefit would using Cyrillic grant English that Latin doesn't? Bearing in mind the failed attempts to reform English spelling over the last 2 centuries.
That's why the Tudor period is particularly opportune; printing is just coming into being in England as the latest thing. So there's a unique opportunity to impose an alphabet change, if someone wants to. There is as yet no big investment in Latin character type, and it is easy to tell a printed book from a hand-scribed one, so the enforcement of a strict rule, or even a lax one, is relatively easy--hand-copied books get a pass for being in Latin characters, but if printed you'd better be able to show your special permit, or point to the page in the volume showing the generic special license given this edition. A tax on Latin character books the Anglo-Cyrillic ones are exempt from is an alternative to simply banning Latin character print books; magic of the marketplace and all that. The King *James Bible is in Cyrillic; so is the *Book of Common Prayer.
The motive to do it is a stretch of course; a move like this has to come down to ideology. But there is precedent going even deeper than what alphabet to use--consider how Iran came to be majority-Shia. The place used to have the usual majority Sunni, Shia minority--maybe significantly more Shia due to holy centers of the Shia denomination being in nearby Iraq sometimes under the Iranian ruling regimes, but Shia was the minority still, albeit larger. Then a Shiite dynasty took over Iran and systematically persecuted Sunni and promoted Shiism--to be sure, Iranians remain subdivided as to exact sect of Shiism; presumably the dynasty wanted a particular one (as I understand it, they go by "number" names--"Fiver," "Sevener," maybe a few others, based on how many rightly guided Imams or whatever the proper title was following Ali each one counts as canonical). But Sunni populations became the small minority. I've even seen some Western historians (thinking of Toynbee here) assume very cynical motives having little to do with deep personal piety of the Padishah--that they anticipated, correctly, that establishing the sectarian difference would fix the boundaries of Iran versus the Ottomans and other neighboring contending powers to be firmer--Sunni powers would have a difficult time holding the Shiite communities, and they would tend to be relatively easy for the Iranian dynasty in power to take back, since they would be the protectors of Shiite Muslims. Vice versa it made it more difficult for Iranian dynasties to take and hold territories outside Iran, but perhaps if the thing was a cynical calculation, the dynasty responsible reasoned that there were natural limits to Iranian reach anyway and they had best consolidate that line and hold it, and forego ambitions beyond that point as too large a stretch for them to sustain. So--Iran is defined to a great degree by this historic and relatively modern (16th-18th century IIRC) move, whether believed to be divinely inspired or an act of Machiavellian statecraft, let better educated historians debate. One way or another, it happened. Similar things have happened elsewhere--Poland used to have quite a few sectarians divergent from Catholicism, but Poland as we know it today, like Ireland, is defined in part by stubborn Catholicism in the face of occupiers who penalized it pretty heavily. In fact I suspect both Polish and Irish Catholic piety have a lot to do with the very fact that adhering to that faith helped define themselves as resisters of the external powers' rule.
A precociously early Reformation with England jumping onto the Protestant bandwagon early, under a Tudor (or if you like some other house emerging from the War of the Roses) monarch aspiring to absolutism (albeit with the support of suitably large and influential social sectors) might take the Machiavellian step of sundering his kingdom from the Continent by imposing a different alphabet; if restricted to print primarily at first (that is, to gain firmer control of the spread of
printed ideology, which could be spammed as it were in much greater volume at a lower cost than hand-copied which would be more like samizdat in its restrictions) it could go over relatively easily. I believe the 16th century represented a period in which literacy rose significantly; jumping in early on that bandwagon might result in masses trained on the Bible to prefer Cyrillic. Very literate people of course would still read Latin and that will doubtless continue to be printed in Roman characters, but the point is the regime controls legitimate printhouses, having a laxer hand with Cyrillic because Russia is very far away and the language is quite alien, while the Latin text stuff, both in Latin language and in various Western European vernaculars is closely scrutinized when imported and regulated when printed in England. Also a preference for Cyrillic is a marker of loyalty and being a team player.
Even with these factors in play, with whoever is the expy for Henry VII being a more activist absolutist and ideologue, it seems likely that there would have to some direct linkage with Russia, a factor I admitted needs some background work to justify. If late Rurikid/Early Romanov, or better yet from my Romanov-despising POV some ATL Muscovite Tsarist dynasty (not necessarily one bit nicer than the Romanovs, so they might as well be precocious Romanovs I suppose) is more powerful on the Western European horizon circa the late 15th century, and England's earlier and more energetic Reformism puts her on a more vigorous stance in Continental matters, conceivably a strong alliance with the Russians might motivate an exchange of royal spouses and other forms of contact. I am backing off of claiming that the Orthodox Church will be a strong influence on emerging Anglican Protestantism, but claiming alliances on paper has been effective in sectarian empire-building before. The Roman Catholics set precedents in the Crusader states--they spurned cooperation with the Orthodox, whose doctrines were pretty close to Latin rite but who politically downplayed the status of the Pope, and set up political "communion" with various other eastern Christian rites that differed much more radically in doctrine and in forms of worship, but were willing to say that they deferred to the Bishop of Rome as the supreme pontiff. The eastern rite people were not expected to change their distinctive doctrines or rites, they were just recognized as more or less equivalently Christian by Roman Catholics, allies not to be persecuted for deviations no matter how hair-raising some doctrines might have seemed to a conventional Latin Catholic. Nor did Roman rites change unless it suited someone in the west to adopt innovations from the east, and surely that was a tightly regulated thing too--saying something was unacceptable in Flanders would not be deemed an insult of the same thing being normal in Tyre or Antioch!
So I suppose in such an ATL there ought to be a bit of Russian flavor to the *Tudors I credit or blame for pushing through this reform in printed writing, which, if persisted in long enough--a century should be enough--would tip the scales of normal literacy in English until people all over Europe just accepted it as part of English identity that they write in that peculiar alphabet, by a few generations from the initial reform, in all forms of writing, handwritten as well as print, because it becomes ubiquitous and normal. They might even take to writing the Latin language in a variant of Anglo-Cyrillic! Meanwhile England, soon to become part of an ATL UK, pursues an increasingly parallel course politically and culturally with OTL. There is a smattering of Russian influence in a few items of vocabulary and a few details of various creed doctrines (mostly in High Church Anglican of course, which is to say they carry over into Episcopalian in America) but some of the Dissenters will carry over a fraction of them too. But by and large a great gulf exists betweeen even Anglicanism and the Russian Orthodox faith--indeed to parallel England politically with OTL there must eventually be a rupture in the Anglo-Russian alliance and both countries go through a long period of mutual hostility. The variations in alphabet will be seized on then by both sides as proving the others are a bunch of barbaric lunatics! Not to mention how funny English sounds to Russian readers or Russian to English ones! Later the OTL cordiality between the Tsarist empire and the USA will seem the more natural, a little bit, because of the mostly shared alphabet, but that relationship will still be peripheral from the point of view of both. To Americans taking over a purchased Alaska, the strange phonetics of Russian monuments and documents will be quite as exotic and quaint as their illegibility to Latin-text-literate OTL Americans was.
I believe I already mentioned and will just mention in passing again various faint interconnections between England and Russia OTL, such as the proposed marriage of Elizabeth to Ivan. I've noticed in modern times a marked tendency for American Episcopalians or British Anglicans to adopt symbolism and perhaps among a substantial minority of the more pious of those denominations, even doctrine, from the Orthodox Church; I have often wondered why.
I am granting that this is quite a stretch, but I think the OP is not utterly frivolous, and furthermore that the period between late 15th and early 17th century, a "long 16th century" as it were, is a cusp where it would be easier to postulate happening than any time before or since. At any rate, if earlier opportunities existed they'd probably butterfly away any chance of parallelism with subsequent OTL developments involving England more firmly. With the *Tudor/Radical Reformation opportunity we can still pretty well slot an England writing in Cyrillic onto a similar trajectory as OTL anyway and get the OP's apparent desire the best. Doing it later would be a more disruptive event. ATL Tudors seem to be the window of best opportunity to me.