Flight Instructor reported an NMAC during approach when another inbound aircraft did not acknowledge ATC instructions and the Tower Controller turned the first aircraft head on to the other.

Narrative

On approach to ZZZ; approach gave my student and I [in] Aircraft X a 130 heading and told us to enter right base XXL. Aircraft Y was behind us and given a heading of 140. We switched to Tower as instructed and Tower instructed us to fly through final for departing traffic and he'd call our turn back. Aircraft Y checked on and since they were now on a collision course with us; was given a 090 heading to fly. They did not read the instruction back; Tower repeated the instruction but was stepped on partially. Aircraft Y asked if that was for them. By this time Tower had turned us back toward the runway and told us to enter left base. This put us head on with Aircraft Y with only 100 feet of separation. I intervened and executed a descending left turn to avoid traffic. Only after this did Tower successfully give instruction to Aircraft Y to turn away fly downwind for XXL.Make sure to listen for your call sign when ATC calls you. If you're not sure if it was for you; ask. Eyes outside especially operating in vicinity of many other aircraft in a busy arrival corridor.