UPDATE 7:44 p.m. with more details throughout. In the wake of middling reviews and a dozy box office, Sting has agreed to join the cast of his first Broadway musical as composer/lyricist. From December 9 through January 10 -- which is to say, for most of the holiday season, when tourists swarm Times Square in search of a Broadway show -- the pop star will replace Jimmy Nail, who plays the foreman of a shipyard on the verge of extinction in the musical, a roman a clef…