C172 flight instructor reported pattern traffic failed to follow clearance which resulted in an NMAC.

Narrative

Student and I were in the traffic pattern as I was teaching him landings. On our last loop in the traffic pattern; a student solo from a flight school in Aircraft Y. Aircraft Y (which will be referred to Aircraft Y) was cleared for takeoff to follow behind us in the left closed traffic. Student pilot read back instructions correctly from the Tower.Around XA:50; I noticed on 6DB's Garmin 375 (GPS/Transponder; also displays traffic) that Aircraft Y had started their turn potentially early. I gave it a quick minute to see if Aircraft Y was actually going to turn ahead of us. A second later the 375 reported that Aircraft Y was at our same altitude; and was turning towards us.At XA:51 several things happened at the same time; I took the controls from the student; executed a 360 degree turn. In the turn I spotted Aircraft Y within roughly a 100 feet at the same altitude; and the 375 alerted for traffic. I alerted the Tower that we were executing a 360 for spacing; and followed behind Aircraft Y. Tower apologized because it had appeared to him that Aircraft Y was going to follow us; and he sneezed at the wrong time. We requested a full stop landing at mid field; due to my lesson almost being completed.Student in Aircraft Y did a full stop; and we followed in behind them. On their full-stop; clear of Runway XX; they requested to taxi back to the ramp. The Tower cleared them '[Taxiway] 1; 2 to the ramp'. We kept a good distance away from Aircraft Y; and they pulled up to [Taxiway] 2; holding short of Runway XX. The Tower Controller asked Aircraft Y if they read back to taxi to the ramp. The student in Aircraft Y confirmed they read back to the ramp. Tower asked if they could make a 180 and return to the ramp. Student said 'unable; trying to complete three full landings and taxi backs for my solo'. Tower sighed and cleared them to takeoff again due to no one else in the pattern.The controller and I have a friendship; afterwards I messaged them and they mentioned they would not have let them takeoff if someone else had been inbound; or in the pattern.Upon shutdown; I noticed the flight school Instructor watching the student. My student went inside to fill out the book; and I approached the instructor to inform him what had happened. Instructor was apologetic; and very thankful I had told him what happened. He felt incredibly bad; mentioned he was lucky I was watching the situation and was thankful it was not worse. He said he would do more training with his student in Aircraft Y before letting them solo again.I believe that the student in Aircraft Y was very good at reading back instructions; however; did not fully know what they meant. He did not have us in sight before turning and almost caused a midair collision. The instructor should not have soloed the student.Sadly; there is not much I think I could have done to prevent this from occurring. I think I did the necessary steps to avoid a collision with Aircraft Y and I am happy with my reaction skills. Potentially I could have radioed and asked Aircraft Y their intentions; but I like how I accomplished things. Aviate; navigate; communicate.