The PA-44 aircraft was cleared to take-off and given a clearance which included the instructions to maintain runway heading and to call Adelaide Approach airborne. At about the same time, the TB20 was approaching Adelaide at 2,000 feet from Parafield to conduct a practice Adelaide/Parafield non-directional beacon (NDB) approach, which uses the Adelaide NDB. The TB20 pilot was cleared to conduct the NDB approach. The PA-44 did not commence its take-off run until about 110 seconds after lining up and being issued with the clearance. During this period, Adelaide Approach East issued a restriction of 2,000 feet for the PA-44, which Tower passed on to the pilot. Once the PA-44 became airborne, Tower advised Approach East that Tower was able to co-ordinate the separation between both aircraft. Approach East then directed Tower to amend the PA-44 clearance to direct to Port Stanvac at 2,000 feet. Tower could not raise the PA-44 as the pilot had changed to Approach radio frequency as instructed. Approach East then reported that he had issued the revised clearance to the PA-44 pilot. Almost a minute passed before Tower advised Approach East that a conflict was rapidly developing between the PA-44 and the TB20. The TB20 pilot, on the outbound leg of the NDB, reported sighting the PA-44 at a distance of about 30 metres, level, on his right and tracking from right to left.