Cessna 172 Flight Instructor reported an NMAC event during training with a non-reporting aircraft. The unannounced aircraft exited the airspace and both aircraft continued safely.

Narrative

I was conducting a training flight for my Certified Flight Instructor-Instrument rating. I was pilot flying; my instructor was pilot monitoring. Using a view limiting device (VLD); we conducted several approaches at our home airport of ZZZ in aircraft C172. I was flying the approaches using the VLD; while my instructor scanned for other traffic. At approximately XA:34; we joined the localizer course for the ILS/LOC XXR approach at ZZZ behind another Cessna. We contacted ZZZ1 tower for transition approval to complete the approach as the Final Approach Fix is positioned within their airspace. ZZZ1 tower approved the transition; advising us of the previously seen Cessna traffic. We noticed another aircraft on our MFD that was crossing the approach course northbound; and ATC advised us of that traffic as well at 1;100ft. At approximately 15:36; as we were about to cross the Final Approach Fix at 1;600ft; we noticed that this aircraft (later identified as Aircraft Y); had entered ZZZ1 airspace; and had begun a very sharp climbing turn to the left and was climbing right at our aircraft. Even with my VLD; I was able to see the sunglasses of the pilot in that aircraft as they suddenly passed 100ft below us. My instructor and I did not have any time for evasive action as the aircraft's turn was sudden and unexpected.Afterwards; I took off the VLD and resumed the approach with my instructor and I both sharing duties for scanning of traffic. ZZZ1 approved frequency change to ZZZ advisory frequency shortly after. We tried contacting Aircraft Y on ZZZ frequency as they were in the immediate vicinity of the airport but were surprised to learn they had not been making any radio calls. The rest of the flight continued without further incident.I believe what caused the problem was Aircraft Y's inadvertent entry into controlled airspace; followed by an evasive turn to exit the Class D immediately; without scanning for other traffic and in doing so; putting themselves on a collision course with our aircraft. ZZZ is an uncontrolled field and while the use of radios is strongly encouraged; there are oftentimes occurrences where aircraft are not using them properly (if at all) and posing hazards to aircraft that are engaged in communication. What can be done to prevent future recurrences may include changing ZZZ to have its own Class D airspace or increasing pilots' awareness of airspace nearby. However it is difficult to determine what else may be accomplished to prevent future recurrences due to inability of determining what the other aircraft's intentions were in this event.