C172 Flight Instructor reported that while on base; there was conflicting traffic reported on final that resulted in taking evasive action contrary to ATC's instructions to avoid a NMAC.

Narrative

Student and Instructor (myself) doing landing practice (touch and goes) at ZZZ in a closed traffic pattern. Clear skies; day VFR; unremarkable wind conditions. Once on the go; ZZZ Tower issued left closed traffic for us and to report left base. Upon turning left base to final for runway XX; just before I could key the mic to report my position; another aircraft called over frequency that they were a 2 and a half mile final for runway XX. Again; before I could transmit my position (not wanting to talk over a transmission) tower asked the other aircraft that was calling to verify their position; in which the aircraft verified 2 mile final for runway XX. Tower issued a command for me to extend downwind. However we were already established on left base. Had we turned right to extend a left downwind; we would have been nose to nose with the other aircraft. And we had not yet made visual contact. I took controls and issued an immediate go-around climb for runway XX making a left climbing turn onto final while immediately sidestepping to the right side of runway centerline to be able to establish visual contact. Once making visual contact; my student ascertained we had approximately 200ft separation. Once clear of traffic; I asked for the tower's phone number to better get an idea of what just transpired. Once I landed; I called tower and they explained that the traffic on final entered the delta [airspace) without establishing two way-radio communication. But not only that; they are believed to have went 'radar contact lost' while talking to ZZZ approach; which at the time; ZZZ approach had been having radar issues which I was not aware of. That aircraft then realized their position and reported themselves in the delta after entering. From what I can gather; the aircraft on short final was potentially on an IFR flight plan (not verified) and was task saturated teaching an instrument student; and had fallen under an expectation bias that ATC would remind them of their position; and thus; tell them to contact tower. If this radio transaction did occur; and approach told them 'radar contact lost'; or 'contact tower'; that aircraft either did not hear them due to mismanagement of their communication panel; approach control did not issue a timely notification to them of their position; and/or they had already opted to switch frequencies due to realizing their position within the Delta airspace and never heard the transmission to them.The decision we made to immediately execute a go around was the correct decision and potentially averted a mid-air collision.