An LAX Controller described a conflict between an arrival issued an uncoordinated go-around and a departure aircraft. The reporter indicated SCT failed to properly coordinate.

Narrative

Aircraft X was on approach for Runway 25L. SoCal TRACON canceled Aircraft X approach clearance because he was losing separation with the traffic in front of him. Training was taking place on Local Control (LC1) and was off frequency coordinating on the land line with Local Control 2 (LC2) for a cross over departure when the approach clearance was canceled by SoCal TRACON. Local Assist (LA1) was also off frequency coordinating. Neither LC1 nor LA1 heard the approach clearance canceled. LC1 cleared Aircraft Y for takeoff from Runway 25R. When LC1 realizes that Aircraft X was on a go around; Aircraft Y was already on the takeoff roll; so LA1 coordinated with SoCal TRACON to see if they were going to break Aircraft Y off the approach. SoCal TRACON said negative and LA1 coordinated to climb Aircraft X to 3;000 FT and Aircraft Y to maintain 2;000 FT. LC1 told Aircraft Y to maintain 2;000 FT and then told Aircraft X to maintain 3;000 FT. Both Aircraft Y and Aircraft X read back the clearance. The instructor at LC1 keyed up and turned Aircraft X to a heading of 200. Aircraft X read back the heading and altitude again. Aircraft Y was offshore heading 235 and maintaining 2;000 FT. Aircraft X did not turn or start his climb and had to be instructed a second time to start his climb and turn left heading 200. Aircraft X once again read back the clearance and climbed to 3;000 on a heading of 200. Aircraft Y was instructed by LC1 to contact SoCal TRACON departure. Aircraft X was then instructed to contact SoCal TRACON departure. When SoCal TRACON cancels an approach they should coordinate with LA to insure that the Tower is aware of the go-around and to let them know if there are any special instructions to be relayed to the aircraft.