22-year-old Dickson joined the Railwaymen last month but failed to make an impact at the club and the Crewe management decided not to extend his contract.
Dickson made made two starts and also came on as a sub against Swindon Town but returned to Charlton last week.
Gillingham chairman Paul Scally told the Gillingham Official Site: "Dickson has super pace and is a bright young player. He will enjoy a good future in the game so it's a great swoop for the team and our super fans."