TRACON Controller described a loss of separation event when coordination and direction to a military GCA Controller was not complied with; noting more specific directions to the GCA Controller would have likely prevented the event.

Narrative

Aircraft X was under the control of the ZZZ GCA (Ground Control Approach) unit making an ILS approach to Runway XX for a practice approach; climb out; and then pick up an IFR on the go. I relayed climb out instructions to the GCA Controller of; 'Turn left heading 360; keep his turn close to the airport; climb and maintain 3;000 FT; and return to my frequency.' A few minutes later; I cleared a C650 for an ILS Runway YY approach into ZZZ1; which is located about 7 miles north/northwest of ZZZ. Seeing that the C650's approach might conflict with the climb out of aircraft X; I contacted ZZZ GCA while aircraft X was approximately 2 mile final and gave them traffic information on the C650. The GCA Controller responded; 'Traffic observed.' About this time; a Handoff Controller was plugging in to assist me at the position and he heard me make this call as I explained to him the traffic situation. About a minute or so later; we observed aircraft X; 2 to 3 miles north of ZZZ at 3;000 FT on a converging course with the C650; about 5 miles east of ZZZ1 also at 3;000 FT; westbound. My Handoff called the GCA to instruct them to turn aircraft X westbound immediately for traffic and also climb it to 4;000 FT. Simultaneously; I called ZZZ1 Tower and issued a traffic alert for the C650 just as the CA alert was going off on my display. Thankfully; the C650 started descending on the approach and aircraft X turned in time to avoid a calamity. Later; I had a phone conversation on a non-recorded line with the GCA Controller. During that conversation; he explained that he had issued the missed approach instructions to aircraft X and then turned control of the aircraft over to the Final Controller. However; it was the Final Controller whom I had issued the traffic information to and who replied that the traffic was observed. I explained to the Controller whom I had on the phone that it was imperative that the other Controller understand that he was accepting responsibility for separating from the exchanged traffic; even if that meant giving the aircraft a stop and go to delay it's climb out; extending its upwind before allowing it to turn to the 360 heading; or requesting an alternate climb out. This was not withstanding the climb out instructions issued by me previously. I explained that he basically should have done anything but what he did! Recommendation; if I had it to do all over again; I would have given very specific suggestions to the GCA Controller on how to make the assigned climb out work; or asked him to advise me if he required a different climb out. Many times these controllers at military installations are not experienced and need to be properly trained on the creative use of control instructions to avoid a suspected conflict.