There has been a wave of script reforms in the 1990s after the Soviet Union had disbanded. Some of that has been reversals of Stalin’s all-Cyrillic policy of the 1940s, which had been preceded by an all-Latin policy for a short while. I think it was Turkmene (or Azerbaijani) that had been switched from Arabic to Latin to Cyrillic back to (slightly different) Latin within 70 years or so. It helps if there wasn’t much literature written in the old script, e.g. because a different, larger language was being used in offices, and it helps if neighboring related languages already use the Latin script.
These were actually all reforms from one alphabetic writing system to the other, of which there are few: Latin, Cyrillic, Greek, Armenian, Georgian and Arabic. The others are more or less syllabic.
Serbian and Croatian or Hindi and Urdu or
Bulgarian Romanian and (Transnistrian) Moldavian are politically claimed to be separate languages, but linguistically really are close dialects of the same language, because they differ almost exclusively by the script used to write them (and the geographic area they’re spoken in).
The Chinese writing system manages to cover a very diverse set of dialects that would be considered separate languages if they used a script with a small number of signs (like an alphabet).
Many of the Indian scripts differ hardly by more than systematic graphic transformations. They have a common ancestor and unified romanization schemes. Still, usually one script is closely tied to one language, although each of them could quite easily be adopted to write neighboring languages. That could provide some benefits.
There have been plans and proposals to convert to Latin from traditional writing systems that are not alphabetic in some countries in the past, e.g. China and Japan, and it has been done in Vietnam, for instance, but usually the syllabic or morphologic elements of the existing native writing system are considered too important to be sacrificed. In most of these cases, there was the matter of cultural identity was also important, because the traditional script had been used for generations. That rarely works in the opposite direction, but it did in Korea where an older native system (Hangul) was revived centuries after its inception to finally (mostly) abolish the foreign (Chinese) system.
If a previously unwritten language gets a writing system applied to it, that’s nowadays usually done based upon the IPA, which is Latin-based. (A notable exception are minority languages spoken in predominantly Islamic areas where the Arabic script is often chosen instead – or alternatively.) During the 19th century, e.g. in North America, there was a trend to invent scripts instead.
Basic knowledge of the Latin script (especially as used in the English writing system) is widespread world-wide (where literacy rates are high enough). So the change to any other script would be more controversial unless it had high cultural value (mostly nationalistic or religious).
There is now a tendency in the “West” to idealize and glorify the “simple life”: Indigenous tribes must be protected from the evil influence of our degenerate industrialized mass-media fast-paced un-culture. That’s just the backlash reverse of prior imperialistic habits where white people brought civilization to the primitives. Therefore, there would be some opposition of well-meaning first-worlders who believe it’s their duty to protect less educated people far away from being taken advantage of by their government, especially if they have reason to think the motive for the change was compatibility with the western world. Unesco may adopt this position in some cases (e.g. Armenia or Georgia), but not in others (e.g. African languages except Ethiopian and Coptic), because the former have unique scripts that contribute to humanity’s global heritage, whereas the latter just adopted one or the other foreign system.