As of 1815 (as per OP request) the Latin alphabet dominated in Western and Central Europe, and (almost totally) in the literate parts of the Western Hemisphere. Cyrillic dominated in the remaining Christian areas of Europe. Both were almost entirely absent elsewhere, except for writing European languages by people from Europe or otherwise Christianized.
In Africa, Afrikaans was mostly written in Latin script (Arabic script was also used). Everywhere else used either a form of Arabic or Ethiopian script (Coptic and its derivates and Tifinagh were very marginal at this point, Vai and other systems did not exist yet) or was largely illitterate. Asia, outside Siberia and the Philippines, almost exclusively used native scripts of various kinds at that point. A few societies had almost no local writing, but whenever writing in a foreign prestige language was used, Latin and Cyrillic was rare except in some parts of the Indonesian archipelago and in some coastal areas of India (where Dutch and Portuguese were used, and English or French to a somewhat lesser extent).
In the Americas, the native pre-Columbian writing tradition was essentially dead, likely beyond plausible recovery. Many peoples has not literacy at all, but almost all who did had access to Latin script almost exclusively (Cyrillic in Alaska, but very marginal). The Cherokee syllabary and the others on that model yet in the very near future.
In non-Ottoman Europe itself, alternatives to Latin and Cyrillic scripts were either long dead, marginal (used by Jewish minorities, or Arabic script used by the Lithuanian Tatars) or moribund (runes) but general literacy was likely past the point when a complete change could be deemed convenient. In Ottoman Europe, Latin was marginal, Cyrillic dominant among non Hellenic Christians. While the Latin dominance was not yet very much established yet, its foundations were. Could it be possible to stem that already rising tide?
I think it requires some radical and unlikely divergences, but I don't think it is impossible.
Here is a tentative sketch. Very unlikely, but here you go:
things in Britain go very badly in the Restoration period. Maybe the last phase of the Maratha Wars goes worse and longer for the Brits, in a way that ramifies into a messy situation both in India and the British Islands, with repercussions all over the Empire. Irish, Scottish and Welsh revivalist nationalisms emerge centering on Celtic languages (perhaps a Cornish cultural movement too) and they opt for a ogham-derived script just for the heck of it, but this leads to a Romantic-radical backlash in England too, where a Populist movement chose a "populist" (phonetic) spelling... and runes to write it, 'cause they are the "native" script after all. This works because there is a revolution going on and the new regime enforces this, while an awful lot of the English people already literate in Latin script are forced to flee. This, however, happens only after the pro-runes movement has had its own generational run in exile, where they plant the notion firmly among the German and Pan-Scandinavian nationalists, who follow a similar path when their own revolutionay movements succeed a little later. Under this influence, many Slavic nationalist movements also opt for promoting "native" literacy: they adapt Glagolitic from scratch instead of established scripts either Latin or Cyrillic (the latter is perceived as too tied to the oppressive Tsardom). All these revolutionaries are very secular, believe in a strange blend of mythic nativism and futuristic progress (compare some forms of right-wing Zionism) and find themselves with enough of a blank slate to work with across Northern, Central and Eastern Europe.
Similarly, in Latin America, maybe as a consequence of the different British attitude, the Criollo revolutionaries take a somewhat different tack... Plata and the Andes actually go for a constitutional monarchy under an Incan noble and chose Quechua as the new "national" language. Spanish does not disappear by any means, but it has competition, and a new script is devised for Quechua and other native languages with official status (you can bet Guarani, Mapudungun and Aymara are on board). This is done to emulate Mexico, where Nahuatl and Yucatec Maya are picked... and a syllabary based on a very haphazard "reconstruction" of the actual Maya glyphs is adopted for both. The reconstruction is so detached from the actual pre-Columbian writing that it does not even matter to discuss if it wrong (it is) but anyway they use that. For now, Italy, France, Spain, Portugal with Brazil and for a while the Netherlands cling to Latin, and Russia, temporarily, to Cyrillic (but only for Russian). Likewise, in the USA English remains written with the Latin script mostly, until the second wave of Caribbean slave revolutions hit the mainland, with their newfound "Liberty script", first invented in Jamaica by former slaves who remembered the Nsibidi signs of South Nigeria and turned it into an actual alphabet under inspiration of English itself. A Black-majority republic using this script warily looks at the rump US (including Canada) across the Potomac and Ohio rivers, but in the US themselves, native languages are increasingly written with derivates of the "Maya" syllabary (also used by Cherokee and others further South).
Disgusted by the "Pagan" runes used by their former colonial masters both English and Dutch, Afrikaners adopt a "holy" ortography - Hebrew. A lot of Protestant denominations in the English and Dutch speaking worlds follow suit, in the ensuing religious fervor, Latin script English loses ground in the US, without disappearing entirely (it also survives in Australia). Proposals to write French in the "Celtic" way are also made, but it would take a new, hardline revolutionary regime to try to enforce that after a long, bloody series of very nationalistic wars in the 1910's and 20's. These same wars also cause a Futurist new regime in Russia embracing neo-Glagolitic, and Italy finally unifying... under a weird sort of ideology that looks back at the Etruscans, and choses to make most Italians literate in a "Italic" script, derived from Etruscan and other non-Latin Italic writing systems, however poorly understood. Yeah, these people are Italian nationalists who hate the Roman legacy (let's say the capital of Italy ITTL is going to be Florence).
The militantly nativist anticlerical movement the soon thereafter creates a new Iberian Union does the same, using a fancyful "Decipherment" of Tartessian to write down "Spanish" (they actually would love to enforce Basque over the whole thing, but this proves too unpractical).
Venezuela still writes Castilian in Latin, and so do South Brazil (the bit that survived the slave revolts) and East Timor with Portuguese, but these are the main holdouts of Latin Script in the world by the year 2010, alongisde English-speaking East Austrialia, Lithuania and Hungary where the local Runicist movements never gained much steam (they had to mark the distance with the Germans somehow; Malta had revived Phoenician in the meantime). Ironically, the only other Germanic language still mostly using Latin script is... Icelandic.
This is a world of deeper nationalism, and where nationalism is more deeply linguistic and even more interested in ancient roots, "pure" origins and all the other myths the Nazis largely discredited ITTL. A world where a philosophical system that resembles what Heidegger did IOTL is dominant paradigm, where "authenticity" is a very serious political concern. Historians and (especially) archaeologists and philologists have MUCH more funding than IOTL (think historical OTL Israel's levels of fascination with archaeology worldwide), though the independent nature of their work is also even less guaranteed, given the politically sensitive nature of their job.
Of course, this would gradually lead to dismantle the notions that created this situations. But by then, they would have shaped the political and cultural context a lot. And there's mass literacy.