First of all, there is this article from NBC (that is also referenced in another answer) which says that Bane's nationality/ethnicity is half British and half Caribbean:
DC Comics describes Bane’s father as a British mercenary and his
mother a rebel from the Caribbean. His life spent in a jail on the
fictional Caribbean Island of Santa Prisca, a seemingly Spanish name.
But they fail to set this into context with a movie franchise that has made extensive changes to the characters of its source material (though they themselves mention some of the changes to his character in the movie, including the casting of a non-Latino actor). For example they speak of a Caribbean prison (which it wasn't in the movie) and the comic character seems to include elements from Talia's movie character, being the child of a mercenary (while Ra's Al Ghul isn't a mercenary in the comics).
So all in all I don't think this article or the comic source material it references can be used as an explanation for the character of Bane as depicted in The Dark Knight Rises, at least not a ono-to-one explanation.
But to come back to the actual question what nationality Bane was then, one could at least use the fact that there is most probably a connection between the comic Bane being half-Carribean and spending his life in a Caribbean prison. So if we set this into context with the movie placing the location of the prison in Morocco (which isn't 100% clear though), it is a very reasonable deduction that the Bane from the movie is indeed to some degree Moroccan (which would also make him look tanned, albeit not after growing up in the pit, though). But I think apart from that we cannot draw any other conclusions, as his backstory is not mentioned in the movie and the comics fail to provide an explanation because of the above mentioned contradictions to the movie.