The adoption of Latin-type letterforms in regions using the Cyrillic script outside of the Russian Empire varied widely depending on the country and context. I will aim to provide a broad overview of when these letterforms were adopted by language.
In Serbia, the linguist Vuk Karadzic reformed the Serbian Cyrillic script to make it more phonetic; the older script was not standardized and quite irregular. In other words, Serbian was written as nonphonetically as English but without standardized spelling. Imagine if I could spell the word "controller" as "kantroler," "kontroller," or "cantroler" with all four spellings being equally valid and none of them directly reflecting pronunciation.
Here's a sample of Serbian Cyrillic prior to Karadzic's reforms (printed in the modern type to highlight its irregular spelling):
Весьма бы мені пріскорбно было, ако бі я кадгод чуо, что ты, мой сыне, упао у пянство, роскошь, безчініе, і непотребное жітіе.
Here's the same sentence in modern Serbian Cyrillic:
Весьма би мени прискорбно било, ако би ја кадгод чуо, да ти, мој сине, упао у пијанство, роскош, безредје, и непотребно животе.
Here's a full page printed in the older Serbian script from a book printed in Venice in 1546:
Karadzic used the new Cyrillic script design to the older form because he felt that it would be accessible to the people since it is easier to write and the letterforms are easier to remember since they are simpler. In addition, several letters found in his script reform (such as J) are not found in the script that preceded it.
The current Serbian alphabet was finalized in 1818 and quickly gained acceptance among the Serbian people, primarily because they appreciated its ease of learning and use when compared to the older script; literacy rates were low because of its difficulty. In addition, a broader cultural and linguistic revival took place in Serbia at the time that sought to create a new national identity as a result of resistance to Ottoman rule. Karadzic and his associates produced as many books in the new script as possible which massively increased its popularity and use. The script was officially adopted in 1868, four years after Karadzic's death but became dominant decades before then.
The process of orthographical reform in Bulgaria began during the period of the Bulgarian National Revival (a period of cultural developments that resisted Ottoman rule in favor of a national identity that was similar to the movement in Serbia at the time). Intellectuals and writers wanted to increase the use of Bulgarian in academia and consequently reformed the language's orthography.
Prior to the Bulgarian orthographic reform, a completely nonstandardized Cyrillic script using non-Latin type letterforms was employed (which was similar to the older one used in Serbia). The number of letters in this script varied widely and it often failed to reflect the pronunciation as a result of its irregularity (different people used different letters for the same sounds) and it failed as a writing system in virtually every way.
However, the adoption of the modern Bulgaria alphabet occurred more slowly than in Serbia and took multiple decades. Some people believed that the new script too closely resembled the Latin script and broke tradition. Another reason for the slower adoption was the relatively small amount of printed literature in the new Bulgarian script which made it difficult for people to practically apply since a single linguist did not lead the production of large amounts of material in the new writing system in a short amount of time, unlike in Serbia.
The use of Latin-shaped letters in Ukrainian originated by the linguist Panteleimon Kulish in the 1860s. Kulish sought to reform the Ukrainian script which was based on the older Cyrillic letter shapes; it was not phonetic and nonstandardized. Different letterforms were used in different regions and people living in the same place sometimes used different spellings for the same word. This made it difficult for people to read and write in their language.
The reformation of Ukrainian orthography occurred during the Ukrainian cultural and linguistic revival which sought to promote and celebrate Ukrainian culture. Ukraine was a part of the Russian Empire at the time and its language had no official recognition or promotion.
Some Ukrainians saw the script as detached from tradition since it altered the letterforms that had been used for centuries. However, people began to accept the new script when Ukrainian writers promoted the use of the new script as part of literature and education. Taras Shevchenko was one of the most major advocates of the new script.
The current Macedonian Cyrillic script using Latin-type Cyrillic letterforms was introduced in 1945. Unlike the other three languages already mentioned, Macedonian was already written phonetically before the introduction of the new script. However, there were multiple standards of the script that often conflicted with each other in terms of spelling. The newer letter shapes were adopted because they are easier to read, write, and learn.
The Belarusian Cyrillic script was introduced in its modern form in 1918. Like the other languages, it was written in many nonstandard forms and was not recognized by the Russian Empire; the modern script was created for use in Soviet schools to stop the suppression of the language.
Other Cyrillic Languages
All other Cyrillic languages were not written in Cyrillic prior to the introduction of the modern Cyrillic script.