Done some googling and can't find much about this.
Did Muslims call their religion Islam in Muhammad's time, or did the term come about later on at some stage?
Was there ever another name for Islam before "Islam" or was there ever another name used by Muslims for their religion in other parts of the world (besides Mohammadism, since that wasn't used by Muslims).
The first use of the term "Islam" appears in the Qur'an. It's used in multiple occasions as the proper name for the Qur'an's religion. A passage of Q 5:4 reads (translation by M. Abdel Haleem):
Today I have perfected your religion for you, completed My blessing upon you, and chosen as your religion islam
Another passage from Q3:19 reads:
True Religion, in God’s eyes, is islam