A Tower Local Control trainee reported they cleared a turbojet for takeoff behind a slower moving aircraft which had began its turn away from the runway. The first aircraft unexpectedly turned back into the path of the jet departure and stopped climbing resulting in a Low Altitude Alert.

Narrative

On local; I departed Aircraft X runway heading 060. Their flight plan had them flying to ZZZZZ intersection which is a heading of 050. Once I switch him to departure at the departure end of the runway Aircraft X appeared to be turning to the left. From my point of view I assumed departure turned the aircraft to the left. Per our SOP once an aircraft turns away from the centerline; the departure controller is not allowed to turn back towards the centerline without approval from local control. The departure controller did not call up and coordinate turning back towards the centerline; regardless if the pilot turned on his own. Once I noticed Aircraft X turning I cleared Aircraft Y for takeoff turning right to a 090 heading. This would provide more than the 15 degree minimum separation required. But as Aircraft Y was on his takeoff roll; Aircraft X started to turn back towards the center line. They ended up being only 2 miles apart without the divergence. I issued traffic to Aircraft Y and insured he turned right to 090. He immediately climbed above Aircraft X and had vertical separation within a few seconds. I believe Aircraft X was having issues because he was still at 1500 ft. 7 miles off the departure end; he also had a low altitude alert at the same time. Anticipating aircraft characteristics he should have climbed higher and faster and it shouldn't have been as close as it ended up being. There was no collision alert for the incident. Suggestion: In the future I will turn the first aircraft to ensure the separation is there and not assume departure had turned the aircraft. Also clear guidance on whether an aircraft turns on his own; if that still requires coordination with local. Since he was supposed to be runway heading is departure allowed to turn back to runway heading without coordination. That could be clearer.