Flight Crew Private; Flight Crew Instrument
Flight Crew Last 90 Days 1; Flight Crew Total 480; Flight Crew Type 408
Communication Breakdown
ATC Issue All Types; Conflict NMAC; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance
Flight Crew Took Evasive Action
Human Factors
During a practice approach RNAV XXL ZZZ on IFR flight plan we were cleared for the approach.  We had also been told to change frequency to the advisory frequency of ZZZ so now not in contact with Approach.  We observed traffic on our 5:30 300 feet below and closing fast.  We were supposed to be descending which would have made the closure rate greater. We attempted to reach out to the other aircraft via the CTAF with out any response and then decide to take evasive maneuvers and turned left away from  traffic.  We made a 360 to avoid the traffic all while in VMC and continued the practice approach and reported the miss to TRACON.  when we switched back to TRACON to report the missed the controller was not happy that we had turned and wanted to know why we said we had to take evasive action for a tragic issue and did so to avoid a collision. He said we did not report it quickly enough and the he could not trust us; yet he did not mention the tragic closing quickly on us before we were handed off in the first place.  We discovered the aircraft behind us and closing on our ADSB display and as soon as we turned away from the direction of travel we saw the aircraft.  If we had followed the approach path exactly we would have descended directly into his path but ahead of him.  We never observed the other aircraft slow down or turn away from the similar heading as us.
M20-K single pilot reported an NMAC event during approach with unreported traffic on a collision course.