It's on the Alan Godfrey 1906 map of Bradford: heading away from the city centre, turn right off Mount Street into Jackson Street, and Royal Street is then the first on the left, running parallel to Mount Street. It's quite short - it only goes as far as Collier Street. (On a modern map Collier Street appears to have been renamed as a continuation of Mulgrave Street, which the old map shows going from Leeds Road to Mount Street and no further.)
Arthur