I’m not sure it’s possible to give a specific date, except to say that the change is very recent - even within the past 50 years. Certainly it dates to the 20th century when scholars started to study Islam from the perspective of Islam itself, rather than looking at it as an exotic foreign thing, an antithesis of Christianity, as had been done in previous centuries.
Here is a brief summary by John Tolan:
...medieval Christian writers did not speak of "Islam" or "Muslims", words unknown (with very few exceptions) in Western languages before the sixteenth century. Instead, Christian writers referred to Muslims by using ethnic terms: Arabs, Turks, Moors, Saracens. Often they called them "Ishmaelites," descendents of the biblical Ishmael, or Hagarenes (from Hagar, Ishmael's mother). Their religion is referred to as the "law of Muhammad" or the "law of the Saracens." (Tolan, Saracens: Islam in the Medieval Imagination, introduction, pg. xvi)
The term “Mohammedan” (or “Mahometan” etc.) is based on the idea that if Christians worship Christ, Muslims must worship Muhammad, and therefore "Mohammedan" is the most appropriate term. This is partly a failure of European imagination (Muhammad was assumed to be like a bizarro Christ - even the literal antichrist), but it was also partly purposeful disrespect and willful ignorance. I answered a previous question about how much medieval Europeans knew about Islam, and the answer was "basically nothing, because they just didn't care". If Islam was what medieval/early modern Christians believed it to be (paganism, or a heretic Christian sect), then there was no point in learning anything about it.
When they did, on rare occasions, study Arabic and Islam, the words “Islam” and “Muslim” were simply interpreted as Arabic words and were translated accordingly. Arabic, the Qur’an, and Islam in general were studied not on their own terms, but so that Christian missionaries could more successfully argue points of doctrine, and ideally ultimately convert them all to Christianity. It wasn’t until the Renaissance and the Reformation that “Islam” and “Muslim” were recognized as more appropriate terms, mostly because Islam was more immediately present in Europe when the Ottomans expanded westward. Use of these terms didn’t really catch on for several more centuries though.
Thomas Jefferson, for example, wrote about “Mahometans” in the 18th century. In a report on the 1880 census of British India, both terms were used:
"Nearly 41 millions are Mohammedans; so that England is by far the greatest Mohammedan power in the world, so that the Queen reigns over about double as many Moslems as the Khalif himself" (quoted in Warren Dockter, Churchill and the Islamic World, pg. 9).
The "Mohammedan question" was raised in various 20th century political issues (what to do about Ottoman territories in the Middle East that were taken over by the British and French, or Muslims in British India, or even Muslims in China). T.E. Lawrence, a champion of the Arab cause, used both “Mohammedan” and “Moslem”. One very late example is from 1971, when H.A.R. Gibb published a book called Mohammedanism. Although he explained that it was no longer the preferred term, he also argued:
"...the term Mohammedan is not in itself unjustified, and in a less self-conscious age Muslims were proud to call their community al-umma al-Muhammadiya" (Gibb, pg. 2)
Gibb was a famous historian of Islam but he's certainly very old-fashioned and out of date now, and he was old fashioned even in 1971. Edward Said spent a lot of time talking about how out-of-touch Gibb (and many others) were. (Said’s Orientalism was first published in 1978.)
In academic terms, people started using “Muslim” instead of “Mohammedan” due to the new (at the time) trend of studying, or at least trying to study Islam from the inside, from the point of view of Muslims and Muslim sources (and not, as Said termed it, from an “orientalist” perspective). This trend was led by Western scholars like Patricia Crone for example; certainly their early attempts at a scholarly interpretation of Islamic history have been challenged since then, but that’s really when academics started looking at Islam critically (in the sense of “source criticism”, rather than just polemic opposition/support).
So the short answer would be, "when people started thinking of Islam as a valid religion and not just an antithesis of Christianity, in the 16th century at the earliest, and in the 1970s at the latest."