An Instructor pilot conducting a practice instrument approach reported a NMAC with opposite direction traffic.

Narrative

We were training for instrument approaches; so I told my student we are going to be doing the practice RNAV XXL approach into ZZZ. In order to shoot this approach we contacted approach to get clearance for this practice approach. They told us to notify when we were established on the approach. I gave my student radar vectors to get established between the initial approach fix of ZZZZZ and the final approach fix of ZZZZZ1; which had a crossing altitude of 3000 feet. My student got established and we got cleared for this approach. We crossed ZZZZZ1 at 3000 and saw Aircraft Y coming directly towards us at a high rate of speed; I began to look for visual contact with the Aircraft Y. Never once did ATC issue a traffic avoidance call for us. By the time we made visual contact with them; they were about 150 feet behind us at our altitude and climbing. We then got traffic alerts from our G1000 system and I; the instructor; took controls and pushed the nose down. After the Aircraft Y passed behind us; we got told traffic no factor; switch to tower frequency. After the switch to tower; we were advised that he was trying to contact us; but we had not been handed over to the tower yet from approach.