The Doctor did have a nickname, as we saw when he met up with an old classmate from the Academy, who addressed him as "Theta Sigma."On a rare occasion when did have to provide a name, he probably went with the old standby of John Smith
I really don't see how that would have worked.
Floz of Gallifrey: "I'm sorry sir, I didn't catch your name."
Doctor: "John Smith."
Floz of Gallifrey: "What kind of name is that?"
Yeah, I know, he could have substituted it with whatever the Gallifreyan version of John Smith is, but I assume he didn't since in my mind that is his real name.