Military flight crew flying a T-45 aircraft reported an NMAC during approach to landing.

Narrative

On an ILS approach to Runway XX we were cleared for a touch and go. We received no further warnings about traffic overhead the field; but I did hear tower talking to traffic south of the field and giving them our position. Tower cleared them to enter left downwind and I assumed it was for Runway XX as well. We opted to take a low approach only and go missed at 200 ft. AGL because my student was flying the aircraft from the aft cockpit. On the climbout Tower said 'I cleared you for a touch and go for traffic.' As the IP (Instructor Pilot); I looked out and saw a light civil passing left to right directly overhead at 2-300 ft. above us in the downwind for Runway XY. I took the controls from the student leveled the jet and proceeded upwind before executing climbout instructions. The Tower called me once on the ground to discuss what evolved and I explained to him that I would have still had a 500 ft. pass with the traffic given the jets typical climbout performance. I also was unaware that the traffic would be overhead and he needed me to remain clear of traffic.