VH-COH was inbound to Townsville in instrument meteorological conditions. The tower instructed the pilot that if he was not visual by 6 miles from the airfield he should continue via the final segment of a Sector "D" DME Arrival Procedure and cleared the aircraft to descend to the minimum allowable altitude. The pilot acknowledged the instruction. At the minima the pilot reported that he was not visual and was instructed to carry out a standard missed approach and climb to 3500 ft. The aircraft was observed to commence a left turn as per the missed approach procedure and the tower then cleared another aircraft, VH-TAV, for takeoff. Just after the VH-TAV became airborne, the tower controller queried the pilot of VH-COV about the altitude and heading of the aircraft. The pilot responded that the aircraft had climbed through 2400 ft and was heading 170 degrees. He was instructed to expediate the turn onto the assigned heading of 290 degrees. VH-TAV was instructed to maintain 1900 ft, to maintain a minimum of 500 ft vertical separation between the aircraft. Within 30 seconds the situation was resolved and the VH-TAV was recleared to its planned altitude. The minimum separtion between the two aircraft was not positively determined but is was probably 2 miles horizontally and in excess of 1000 ft vertically. The pilot of VH-COV did not have the Townsville DME Arrival Chart open in front of him during the final stages of the approach as he was not expecting to be instructed to carry out that type of approach, and he believed he would become visual before reaching the airfield. However, he acknowledged the missed approach instruction and did not request guidance. During the missed approach some of the passengers reported to the pilot that they could see the airfield and it is possible that the pilot allowed his attention to be diverted and incorrectly turned the aircraft to the right.