C172 Flight Instructor reported pattern traffic failed to communicate which resulted in a NMAC

Narrative

I was instructing on an IFR flight plan for an instrument student. We were in Aircraft X cleared for the straight in instrument approach to ZZZ Rwy XX. Traffic was in pattern for rwy XX. I called 10 mile final; 6; 4; 3; 2.5; and 1.5 mile final calls trying to get other aircraft to acknowledge as it looked like our paths might be converging with a standard pattern. Lost visual with other aircraft momentarily before it apparently turned base. The base aircraft was Aircraft Y. Aircraft made zero calls in the pattern and turned base and continued directly on a collision course with us. Upon regaining visual to aircraft (about 500 feet left of us; same altitude); immediately took controls from student; full power; climbing right turn away from aircraft and executed missed once safe distance had been established. Estimating roughly 500 feet as I have never been that close to another aircraft in flight; and his course was coming into us; may have been slightly further or closer. After calling missed and climbing away; made radio call that they had not made any calls and almost hit us. Other aircraft (Aircraft Y) responded that they called final; indicating that they did in fact have a radio. A third aircraft in the pattern confirmed they had 'made no calls in the pattern and almost flew right into that guy'. Aircraft Y did not respond.