In Vietnam, Chinese was the language of government, scholarship, and literature into the 20th century. This was due to the long periods of Chinese control over Vietnamese speaking peoples and their proximity to each other, as well as Vietnamese speakers' affinity for Chinese culture and state ideology. Chinese rule over the area that today includes Hanoi persisted into the eleventh century, bringing Chinese speakers into the area who used a regional version of the language. Meanwhile, the non-Chinese speaking peoples here spoke a different language belonging to the Mon-Khmer languages that prevailed in areas including today's Burma, Cambodia, and southern Vietnam. But many Chinese words were adopted into this proto Vietnamese Mon-Khmer language, and we can assume that many people were bilingual. When Chinese rule ended, the Chinese speaking population integrated into this Mon-Khmer language, bringing even more Chinese words with them, and giving rise to what to today's Vietnamese. It was at this time, the twelfth and thirteenth centuries, that Vietnamese speakers first wrote in vernacular Vietnamese (chữ Nôm).

For an explanation of chữ Nôm, see the example provided by u/keyilan here. Essentially Nôm combined aspects of different Chinese characters to represent both meaning and pronunciation, or simply took a Chinese character for its similar pronunciation in Vietnamese. And because much of the vocabulary is derived from Chinese, words could be represented with their Chinese characters. The problem, as you can see, is that teaching Nôm to Vietnamese students required knowledge of Chinese characters. And so in the past, people joked that Nôm was simply something that retired mandarins and scholars used for their own amusement. Chinese remained the written language of the elite. [Even into the nineteenth century, some mandarins spoke only Chinese and relied on interpreters to deal with Vietnamese speakers.]

In response, during the mid-17th century, Vietnamese began using Nôm as a pedagogical tool to aid students learning literary Chinese. After all, if you wanted to pass the Confucian style imperial examinations, write poetry, or engage with the classic Chinese works that remained the basis of scholarship and government, you needed to learn Chinese, not just Nôm. For this reason the Chinese script (literally chữ Hán) was also called chữ Nho (Confucian script) by Vietnamese.

Things did change under the short lived Tây Sơn insurgent dynasty (~1789-1802). For much of the early modern history, Vietnamese speakers were locked in a civil war between two rival states, the Nguyễn and Trịnh dynasties. The two states were divided on either side of the 16th parallel, with the Trịnh ruling what is today northern Vietnam, while the Nguyễn expanded southward occupying what would later be called Annam or central Vietnam. A trio of brothers led the Tây Sơn rebellion. The most powerful of the brothers was a tax collector who likely was familiar with Chinese, but used Nôm for daily practices. As opposed to the Confucian scholars in the Nguyễn and Trịnh courts, our best guess is that the Tây Sơn conducted their business in Nôm. When the Tây Sơn successfully defeated bother the southern Nguyễn and northern Trịnh, they changed the official state written language from Chinese to Nôm.

As a result, a resurgence of literary works written in Nôm occurred in the 1790s. The Tale of Kieu is considered the masterpiece of Vietnamese literature, written by Nguyễn Du after he led an official delegation to Beijing in 1813. But The Tale of Kieu is not an original work. It's Nguyễn Du's translation, very well done, of a Chinese novel he read during this trip into chữ Nôm. I mention this to highlight that knowing Chinese, and Chinese culture, was at the core of the Vietnamese elite identity. Part of the reason that French administrators began investing in institutions of higher education and promoting the Latin script of Vietnamese (quốc ngữ) in the early 1900s was the threat presented by the continued influence of China.

Going back a bit, in the 1600s Catholic missionaries hoping to convert Vietnamese speakers devised quốc ngữ, a Latin script with diacritical and tonal markings that represent the specific sounds of the Vietnamese languages. For the next two hundred years it remained more an ecclesiastical translation tool limited to the Catholic community. When French naval forces began to conquer southern Vietnam in the 1850s, the need for translators and language instruction grew quickly. French officers were at times limited to communicating with Vietnamese (often Catholic) counterparts via the basic Latin they learned in grade school! The head of the new translators school in Saigon, Pétrus Trương Vĩnh Kỳ would become the strongest advocate of quốc ngữ. By the 1860s he and others were widely publishing books and newspapers in quốc ngữ.

French military administrators disagreed whether they should teach quốc ngữ and French, or eliminate quốc ngữ and enforce French as the official language. This was a reflection of the inherent schizophrenia of French colonial rule, which vacillated between policies of direct and indirect colonialism, between assimilation of the colonial peoples into French culture or association between Vietnam and France. That contradiction was never resolved. Southern Vietnam became a French colony while central and northern Vietnam became protectorates, theoretically independent, but increasingly under direct French rule. In practice, quốc ngữ operated like chữ Nôm had in the learning of Chinese. It was much easier for students to learn this Latin script to write Vietnamese in tandem with the study French. Nôm remained a difficult script to learn, limited to a small elite, and required knowledge of Chinese as well -- and for French colonialists China was a backwards culture to be avoided and overcome.

By the first years of 1900 French sponsored schools in northern Vietnam had also begun using quốc ngữ in primary and secondary education. But for elites, Chinese and Eastern culture remained a dominant force. It was troubling that Vietnamese read Chinese translations of western works, but more so, it was downright dangerous when they looked to Sun Yat-sen's republican revolution or Japan's modernizing empire. They were supposed to admire French thinkers and their colonial project! When the famous revolutionaries Hồ Chí Minh and Phan Bội Châu were writing letters to each other in the 1920s, they were writing in Chinese characters. French administrators were convinced. They increased their support for secondary schools, scholarships to study in France, and subventions to promote the publication of French works translated into quốc ngữ. In the last days of 1918 the emperor Khải Định, a firm believer in colonial modernity who would send his son to study in France for a decade, decreed an end to the imperial exams. The Confucian examinations were over now, and administration done more and more through French and Vietnamese. Increasingly Chữ Nôm had little practical application.

The expansion of Vietnam's urban education and print culture, including novels and journals in quốc ngữ, spurred the growth of the Latin script during the 1920-40s. Of course the colonial project was invested too in producing more native bureaucrats who could use this new script and translate documents easily between French and Vietnamese. Quốc ngữ's ease of learning allowed literacy to spread in a way that chữ Nôm could not have. By the 1940s, plenty of elites could still use chữ Nôm and Chinese, but it was only in the pursuit of classical poetry rather than in government or scholarly writing.