Doesn't actually seem to be an "underwater launch". We have close-ups on the rear body of the missile on a countdown to launch. It has a single vertical fin and I believe we can see boosters at the side of the rear body. There is also a dorsal trunking line. It launches close to the surface, and I am unsure if it is at sea or land, though the horizon suggests sea. During the violent manouvre which leave the craft facing 180 degrees opposite the launch direction we can see swept wings just a little over halfway down the fuselage.
I think this is probably one of the Teledyne-Ryan family of drones. Andreas' site has a nice graphic of the family here. They look pretty close to the craft in the video. http://www.designation-systems.net/dusrm/m-34.html