I don't think an early POD would help much here. The Greeks had their alphabet, they considered their literature and culture to be superior to Latin, and to be blunt, they had a point there. It is all but impossible to see Western missionaries successful enough to literally take over all of Eastern Europe. And if we posit that the Slavic civilisationms do not take their writing system from the Greeks, they had other candidates to adapt closer to home.
I think it more likely for that shift to happen in the nineteenth or twentieth century in a massive social engineering push. Atatürk's government did it in Turkey (against strong opposition, but nonetheless, they did it). Of course a Cyrillic alphabet is much more practical for Slavic languages, but then, the Latin alphabet isn't exactly ideal for Germanic ones, either (let's not talk about the abomination that is phoneme-grapheme correspondence in English). It doesn't matter overmuch in the end, and it wouldn't for Slavic tongues, either. The Poles and Czechs do fine with a Latin alphabet for their languages. If a government wanted to push this through hard enough, it would not be too difficult. Have it precede or conincide with mass literacy drives and you are there. I'm not sure that Peter the Great is a likely candidate - he would more likely make the entire nobility switch to a foreign language than teach the Russian people a different writing system. But if we could find a POD involving him, that would create an example for almost every other Eastern European nation to emulate come the end of Ottoman dominance. Especially if politics could spin adherence to the Church Slavic alphabet as 'collaborationist'.