Each of these words means "according to a statement or affirmation," but the two have slightly different connotations."The driver was allegedly ill" means that someone asserted, claimed, or openly stated an opinion of the driver's condition."The driver was reportedly ill" means that the driver's condition was actually reported in writing or through rumor, gossip, or common talk."The driver was reputedly ill" indicates that he was estimated to be or considered ill, a condition not necessarily made public or openly declared. Reputedly also has a slight connection with reputation, a shade of meaning not present in allegedly or reportedly.
Dictionary of problem words and expressions. Harry Shaw. 1975.