The American cut of the movie has an intro narrated by Joseph Cotton, who played Holly Martins. The wording might differ (since the movie is clearly Holly's first time in Vienna)
Yeah, I'm talking about the version (which is even on my US DVD) where the narrator is some black marketeer neither Martins nor Lime. "I never knew the old Vienna before the war with its Strauss music, its glamour and easy charm. Constantinople suited me better. I really got to know it in the classic period of the Black Market. We'd run anything if people wanted it enough - mmm - had the money to pay. Of course, a situation like that does tempt amateurs but you know they can't stay the course like a professional."
For some reason, I always assumed the British narration was Major Calloway (Trevor Howard), but now that you quote it, I realize it can't be Calloway. Interesting! The narrator must be someone we never meet.