I don't know if the locals prefer to call their town Bombay or Mumbai. I'm willing to assume they prefer the latter. For a few centuries, however, the English name was Bombay, until sometime in the 1970s or 1980s, when someone (who?) decided this was a Colonial name, and disrespectful of the natives, so we all traipsed over to Mumbai. Or Beijing, or what have you.
But notice, this consideration is offered only to some natives, not to others. It would never occur to anyone that we must say Warszawa rather than Warsaw, Praha rather than Prague, and interestingly, not al-Halil rather than Hebron. Not to mention Jerusalem, which was called Yerushalayim while the distant forbears of the English speakers were still camping in forest clearings, nor el-Quds, the name given to the same town about the time the English language was still half a millenium away.