For a long time, England had a reputation as an uncultured backwater, and the English language likewise was perceived as barbaric and uncouth. (Partly this is due to geography-- when Rome was the center of the cultured world, England was perceived as peripheral). But starting in the 16th century, the reputation of England and English along with it began to change.

As early as the 14th century, there had been a wider movement throughout Europe that legitimized vernacular languages and literatures, and England was no different, even if these changes came about more slowly there. (Think Dante and Boccaccio, who were roughly contemporary with Chaucer). But Latin was still the lingua franca of the educated elite, particularly in universities. A number of technological, literary, and philosophical developments contributed to the perception of English as a learned language.

First, with regard to technology, the printing press was instrumental in regularizing the English language and spreading its influence. Even Shakespeare never spelled his name the same way twice (at least in the fragments of his handwriting we have). Also thanks to the printing press, literacy rates begin to rise thanks to the wider availability of texts.

Second, literature has a big effect on the perception of English as a legitimate, cultured language. Putnam's Art of English Poesy (among other works defending literature and other works teaching grammatical rules in English) argues for English's comparability to Latin in terms of grammatical rules and rhetorical techniques. (Forcing English to conform to the rules of Latin grammar is where we get the notion that you should never split infinitives-- because in Latin they're just one word). In 1598, Gabriel Harvey, a Cambridge professor, wrote a note at the end of his copy of Chaucer's works listing the luminaries of English literature, which included Chaucer, Shakespeare, and others. He describes Hamlet as material worthy of "the wiser sort." The fact that a professor of rhetoric considers works written in English worthy of scholarly attention indicates that the reputation of the English language is starting to change.

But I think perhaps the most significant development that would contribute to university education being conducted in English is the Scientific Revolution. In the early 17th century, Francis Bacon lays out a plan for the reform of the sciences that ends up ushering in a preference for plain style prose (itself a stylistic trend coming out of the Protestant Reformation). Part of this plainness means communicating in the vernacular. (Set aside for the moment that Bacon himself wrote extensively in Latin, in part because he thought that the unwashed masses should not be privy to the secrets of nature). By the end of the 17th century, the Royal Society is founded on Bacon's principles for clear, direct communication of empirical observations, which I think contributes significantly to a shift toward English as the primary language of the educated elite.

Of course, Latin still continues to be part of University education, but thanks to England's growing reputation as a colonial power and the trend for communicating philosophical/scientific material in the vernacular, a great deal of education is communicated in English by this period.