Flight Instructor on training flight with student reported NMAC with another aircraft in airport traffic pattern.

Narrative

While on a training flight with a student practicing pattern work at a nearby local airport; we were involved in a near miss requiring evasive action to be taken. After 5 or so laps in a busy pattern making left traffic; the traffic load reduced so I requested right traffic for training as requested by the students primary instructor. Tower granted the request and assigned right traffic. After our previous touch and go; an archer that had been holding short for an East departure VFR had been cleared for takeoff. We leveled at 1500 ft. MSL in the downwind just abeam the departure end. The student points out to the right to bring a potential collision hazard to my attention from the departing Aircraft Y who was now climbing directly at as from our 2 o'clock. They were climbing on a direct collision course with us. As I made visual contact with them I hear Tower calling them saying 'NXXXX what are you doing you're cutting people off in the pattern; I said I would call your turn on course' during this time I called for the controls and started an immediate and aggressive full power climb. Immediately following the previous transmission Tower calls us to provide us with the traffic alert. I responded that I had the traffic in sight and they were passing under us now. The Aircraft Y apologized on frequency and stated they thought they had heard Tower grant them their turn on course at the time of their takeoff clearance. Contributing factors are that the airport does not have radar and handles a large volume of traffic from multiple flight schools from different airports in the area. The Tower does a fantastic job at handling traffic without use of radar; often even better than other nearby airports who have the use of radar. This airport needs radar!