Small Aircraft; Low Wing; 1 Eng; Fixed Gear
Flight Crew Commercial; Flight Crew Flight Instructor; Flight Crew Instrument; Flight Crew Multiengine
Flight Crew Last 90 Days 200; Flight Crew Total 3500; Flight Crew Type 2000
Communication Breakdown; Distraction; Situational Awareness; Training / Qualification; Workload; Confusion
Conflict NMAC; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Published Material / Policy
Flight Crew Executed Go Around / Missed Approach; Flight Crew Took Evasive Action
Environment - Non Weather Related; Aircraft; Staffing; Human Factors
Aircraft X was conducting IFR approach training in VFR conditions. I was the CFI and I was providing separation and sequencing for traffic while monitoring the student pilot (who was simulated IFR).We were authorized and cleared by Approach to conduct the LDA-XXR approach into ZZZ.  Approach vectored us onto several headings as well as advising us they were vectoring us through the final approach course for sequencing of additional aircraft at ZZZ. Approach had just cleared another aircraft ahead of us into ZZZ via the RNAV- XXR.I obtained ATIS at ZZZ which reported Runway XX in use and wind 280 @10; 10 SM visibility.Approach cleared us with a vector heading to intercept Final Approach course into ZZZ; while at the same time advising us ZZZ Tower was now closed and we should maintain VFR; and separation or advisory service was cancelled. We were instructed to switch to ZZZ CTAF.When we switched to ZZZ CTAF; we noted that there were several aircraft in the VFR traffic pattern for Runway XXL.  We made advisory traffic calls indicating we were inbound for a practice LDA-XXR approach and would be conducting the published missed approach with a right hand turn out back to the ZZZ VOR.  We made at least (3) of these advisory calls.I told my student to expect us to sidestep to the right of Runway XXR when we reached the final approach fix (ZZZ-VOR) and we would not descend below 900 feet. (Final approach minimum descent for straight in approach is 780 feet.) My student misunderstood the 'sidestep' instruction and during this confusion we were late to make the sidestep maneuver and we unintentionally descended to 800 feet.Just as we crossed the (ZZZ-VOR) I saw an aircraft headed directly toward us (approximate same altitude) and directly over Runway XXR apparently on downwind in a right traffic pattern near the approach end of Runway YYL for Runway XXL at ZZZ.I also heard an aircraft advising that they would yield to the inbound LDA traffic to assist in separation. I was perplexed that the existing traffic appeared to be conducting right hand traffic for XXL since the FAA Airports Directory advises left hand traffic for XXL.I decided that any further pursuit to complete this LDA-XXR approach was no longer prudent and I instructed my student to immediately initiate the published missed approach (right hand turn) and simultaneously I heard an unknown aircraft angrily advising we overflew his aircraft with less than 200 feet separation. I assume that aircraft was turning on to right base for runway XXL. I did not see that aircraft.  However; I could still see an aircraft directly in front of me and directly over Runway XXR about midfield and at approximate same altitude.There were several heated exchanges of radio traffic and someone demanded ATC traffic tapes be pulled to review this event. Whoever reported the 200 feet near miss also concluded that we should never have even attempted the LDA-XX approach in the first place because sequencing could not be successfully accomplished. In my opinion; this circumstance/event became dangerously compromised because of the unusual ZZZ closed Tower operations and the infrequent non-towered airport operations by VFR traffic during daylight conditions at ZZZ. I also believe the recent closure of the tower today (at XA00 local) caused some confusion by all who were conducting work in the traffic pattern.I will not conduct any further IFR practice training approaches at ZZZ as long as the ZZZ Tower is closed and non-functional. I highly recommend that ATC advise no further practice approaches will be conducted at ZZZ during the closed tower hours.
Instructor pilot reported a NMAC with an aircraft on final at an airport with the Tower closed at the time of the event.