The reason Widge hesitates to leave with the play book is the same reason he hesitated to continue writing other preachers sermons down for Dr. Bright to later deliver as his own sermons. Widge's reason has nothing to do with a sense of right or wrong. A an apprentice he wasn't given moral instruction and nor was he given a sense of loyalty. He was told instead that his life and actions were owned by his master (Dr. Bright, then Simon Bass) and that he must do as he is told to do mush as a donkey or a hunting dog would do. At first the reason Widge hesitated to leave with the play book is that he feared getting caught. This is later compounded by the new found sense of community and friendship that was beginning to rise up for Widge because of the communal environment that was part of an acting apprentice's life.
Personally, I think it's because he needed to go at the exact correct timing which should of been then... but I think he was a little scared to leave then Falconer coulda got upset at the fact he's late! & it's not in charectery. Eh, maybe he just liked the play business.