Flight Instructor reported they failed to take control of the aircraft when the student did not make a radio call and climbed into traffic which resulted in a NMAC.

Narrative

Doing an opposite direction practice approach LOC 17 into SWO with a student under the hood. Student got behind the airplane past the FAF and ended up calling up on a 1.5 mile final instead of at the FAF. Tower had to scramble to get us off the approach corridor and side-step into a left downwind for Runway 35. I had the student take off the hood to continue to land. We were staying at the circling approach minimums which were 1480 MSL. There was inbound traffic on a left base for Runway 35 that was 2 miles out. By the time we got to the runway threshold; Tower told us to make a right 360 for sequencing as we were number 2 and the other plane was number 1. The student started to make a 360 as I was looking out the window for the traffic. I didn't realize that the plane had climbed 300 feet during the turn. Right as I saw the traffic directly in front of us; the TCAS went off saying 'traffic; same altitude 100'' and I looked at the altimeter noting the change in altitude. I told my student to keep climbing to avoid the traffic. He leveled off at 2200 and kept the circle coming to rejoin the left downwind for Runway 35. What I should have done was taken the controls as soon as I saw the deviation in altitude. By staying at 500 ft. we would have easily avoided the traffic and it wouldn't have been a factor. The student was struggling to maintain the approach and was descending at 1500 FPM; definitely not a stable or normal descent to land. My inaction during the approach to ask Tower to break off the unstable approach led to the degradation of situational awareness. My focus was inside the airplane trying to help the student regain control of the descent rather than listening to the busy radio calls surrounding the airport.