C172 flight instructor reported the Tower Controllers instructions for sequencing in the VFR pattern caused them to have an NMAC with another aircraft in the pattern.

Narrative

A loss of separation occurred between a Aircraft X and Aircraft Y in the traffic pattern at ZZZ on Date at approximately XA:23pm local time. I was the CFI in Aircraft X performing a dual training flight with another CFI applicant. Upon entering the right downwind for Runway XX at ZZZ; we were instructed to make two left 360s for sequencing; one at midfield and an extended right downwind. After completing the second left 360; the Controller instructed us to rejoin the right downwind at midfield. This required us to fly the opposite direction on the downwind for about two minutes. During that time Aircraft Y checked on and was instructed to enter midfield right downwind for Runway XX. As we were proceeding to midfield; I noted opposite direction traffic (Aircraft Y) converging on the right downwind. I asked my student to widen out to the right slightly to pass north of the aircraft on the downwind; as we had traffic in sight. When we were just about to pass abeam one another; the controller realized her error and issued a traffic alert to Aircraft Y. As Aircraft Y was answering; the controller told Aircraft Y to make a left 360; blocking their readback. She issued the instruction again. Aircraft Y turned directly toward us on the downwind. The Controller then directed us to turn left and rejoin the downwind. The Controller continued to talk over other aircraft resulting in a couple blocked transmissions. I saw Aircraft Y turn directly toward us; so I took control of the aircraft and entered a 55-degree bank to the left to avoid the other aircraft. I would estimate that we passed each other by about 200 ft. at the same altitude. I believe the near miss resulted from a lack of situational awareness by the Controller.