Flight Instructor reported a NMAC at a non-towered airport during approach.

Narrative

While instructing an instrument student we were shooting the RNAVXX circling to land XYApproach into ZZZ. While in the procedure turn another plane departed runway XY making the call 'departing XY setting up for RNAV XX'. I had my student continue the approach thinking departing traffic would depart to either Right or Left to get 'set up' for approach. However they continued straight out flying direct to ZZZZZ; while my student is continuing to make calls. I had instructed the student to make calls more often (15NM;12NM;10NM) while excessive I wanted departing traffic to know where we were at on the approach. When I saw departing plane continuing direct to ZZZZZ. I asked on CTAF if he was planning on over flying us. After not hearing a response. I believe they made one but was stepped on by other aircraft. MFD (Multi-function Flight Display) gave off traffic alert and I took controls and diverted to the right. At time of diversion MFD said traffic was +04 above. I made call that I was diverting to the right of final approach course and vectored student back on final approach course. Talked to departing pilot he said that his MFD showed +10 he as well made a call after my diversion call saying he was '1000 above and I should've continued approach. Both pilots and airplanes are from the same flight school both pilots have talked to maintenance for ADSB altitude on the MFD being so conflicting.As a flight instructor I could've had my student divert earlier as to avoid any potential collision.