How and when did the term 'Mohammedans' evolve into 'Muslims?

They're completely separate terms. One didn't evolve into the other. The key reason why "Mohammedan" is not just considered the English word for Muslim is explained in the Oxford English Dictionary: > As was noted in Eng. Today (1992) Apr. read more

The ones that hated them said: “Meh, you’re not Muslims, you’re Muhammadans”. But the Muslims again said: “No, we are named Muslims” (similar to what Muhammad Ali did in a box match vs Ernie Terrell). And voilà, the term “Mohammedans” evolved into “Muslims”. read more

The usage of the ‘Muhammedan’ word was made necessary by Christian visitors coming into contact with Asiatic & African nations that followed the ‘creed of Mohammad’ but did not fall into the 2–3 ‘races’ abutting the south and east of Europe, and hence were in no sense Turkish or Moorish (Persians, Indians and Arabs chief among these). read more