Indeed, church Latin had different pronunciations in different regions; some musicologists are very fond of trying to get Spanish Renaissance polyphony to sound like the "right kind" of Latin. Which I suppose counts as "putting on" an accent, although to be fair most people probably don't have a natural accent for Latin!

Ironically, because I was a choirboy in Liverpool and thus learned to sing in Latin surrounded by other choirboys in Liverpool, when I sing Latin I sound more Scouse than when I sing a lot of stuff in English.