During OJT at a TRACON; a potential conflict occurred when traffic was released off opposing runways from two airports with weather complications and late radio contact by one aircraft involved.

Narrative

I was providing OJT. Due to weather in the area; non-standard departure headings were required. ZZZ and ZZZ1 are about 11 miles due east/west of each other. We had a T45 released off ZZZ on a 130 heading and a P3 released off ZZZ1 on a 310 heading. This would put these aircraft on slightly converging courses. We had anticipated turning the P3 northbound as soon as they called; [but] the P3 didn't call until they were nine miles from the airport. We tried to get the P3 on frequency quickly by coordinating with ZZZ1 Tower; but he was lost in 'frequency land.' The T45 was very slow to climb also; resulting in a converging situation. The trainee was slow to react correctly. I took over the position and turned the T45 southbound away from the P3 and had them expedite their climb. As the T45 was turning and climbing he reported the P3 in sight and I told him to maintain visual separation from the P3. The conflict alert did not activate. I think they passed two to two and one half miles apart and about 800 to 900 FT. Recommend not releasing aircraft on opposing headings. We were trying to expedite traffic and keep things flowing smoothly. This was good for the trainee to see how weather complicates things.