An early POD would be the Czech and Moravian lands cleaving to the heritage of SS Cyril and Methodius (which would have required resisting some serious pressure from the West) and managing to spread that influence to Poland so that they all remain in the Eastern Orthodox camp after the Great Schism. The Slavonic Mass pretty much guarantees the chancery and lay use of Cyrillic.
The POD does seem plausible, but it would take much to achieve it. Having Louis the German's expedition against Moravia in 864 fail may benefit Rastislav's position. Carloman succeeding in his revolt in 862, during which he allied Rastislav, may also benefit Great Moravia, though in what way I'm not sure. If that doesn't work out, having Carloman's and Charles the Fat's engagements in 868 fail as well would help. Eliminating Svatopluk, Rastislav's nephew, would also possibly work, for he ended up entering in negotiations with Carloman without Rastislav's knowledge, and accepted Carloman's lordship over his person and his realm, thus setting himself up as his successor; so killing him off either before he does that or having him not be warned over the conspiracy to kill him, leading to his assassination, would work out well.
If we assume Rastislav's rule stays stable enough and successful enough against the Franks, then he'd more than likely have his son (which we could assume he had, possibly counting Bořivoj and Gorazd, albeit unconfirmed) succeed him. Unlike with Svatopluk, we wouldn't see a conspiracy form against Methodius and his disciples, preventing the promotion of the Swabian priest Witching as Bishop of Nitra, and thus, after Methodius' death in 885, preventing the banishment of Methodius' disciples and those educated at the Great Moravian Academy founded by Cyril. Since there is no conspiracy influencing the Pope's views and opinions on what's going on with the Archdiocese in the region, as long as Methodius' successor continues to toe the line he did, then there shouldn't be any moves made religiously. Assuming things continue developing quite dandy, those educated at the Academy may head north towards Poland and help in TTL's Baptism of Poland, and since they'd be against the German clergy, they'd more than likely follow the example of the Moravians.
The side-effect of this means that TTL!Cyrillic develops in Moravia or Poland rather than Bulgaria. Though linguistically things weren't much different yet, you know how those butterflies work.
Another one is no unions between Lithuania and Poland - either Lithuania gets swallowed by Muscovy, or Lithuania's chancery Slavonic (written in Cyrillic) remains its only prestigious language, so that when Lithuanian gets codified, the use of Cyrillic seems self-evident.
Lithuanian Cyrillic based on Chancery Slavonic does sound fascinating, lending itself to similarities with Belorussian and Ukrainian, since they derive from it too. Having Lithuania be swallowed by Muscovy would probably push its Cyrillic to develop more akin to Belorussian, gaining more and more Russian elements as the years went by.
If the POD is happening in the modern times, I can imagine the Russian powers cracking down on the Poles and Lithuanians even harder than they did IOTL after the 1863 uprising, and succeed in imposing the use of Cyrillic (AKA 'grażdanka') in the few Polish and Lithuanian books they allow to print (attempts at that were made IOTL) and eradicating the book smugglers. If they are content with the results, they might start doing the same with the other non-Russian subjects of the Empire, except for the ethnic Germans (and even then, if relations with Germany or Austria went sour, they might be tempted to try). After all, the Soviets did pretty much the same, albeit on a slightly smaller scale.
At the first glance, the survival of the Russian Empire seems like the best way to go about it. On the other hand, the empire was generally content to let Muslim minorities use their Arabic-based alphabets, so the transition of groups of Kazakhs to Cyrillic may actually be slower compared to OTL. And Mongolia might not be converted to Cyrillic.
So is there any way to offset those (possible) losses? Well, maybe if the Russian Empire doesn't merely survive, but thrives. Stronger Russian influence over the Romanian principality might have prevented its switch from Cyrillic to Latin...that's around 20 million people, potentially. And there was a project by a group of Kurdish intellectuals to develop a Cyrillic alphabet for Kurds, in the last few years before WWI. Having a nation of over 30 million adopt Cyrillic would be a massive gain for the script (if the project succeeds, anyway, which is admittedly a big "if").
Though I'm not sure how much stronger Russian influence would be needed to prevent Romanian's shift from Cyrillic to Latin, everything else suggested in relation to the Russian Empire could possibly work! Kurdish Cyrillic would be a neat thing to see.