DA20 Flight Instructor with student reported the nose wheel separated from the aircraft during takeoff roll. Flight crew continued climbout and returned to their home airport where upon landing the nose gear collapsed.

Narrative

I walked out to the aircraft ramp at the ZZZ1 Airport to conduct the flight portion of a Private Pilot mock check-ride stage check. The student pilot was involved in his preflight check. Once he finished; I conducted my own walk around and observed no obvious abnormalities with the aircraft. The weather was reported as calm winds; 10 SM and no ceiling. The student pilot performed his cockpit checks and engine start. We then taxied to the approach end of Runway X and the student conducted the engine run up and pre-takeoff checks; again no abnormalities. He taxied to the threshold line of Runway X and initiated a short field takeoff as we departed on a simulated cross country to ZZZ2. While enroute; I gave the student pilot a scenario that required him to divert to ZZZ. He entered the left-hand traffic pattern for Runway XX and made all the appropriate radio calls. I asked for a short field landing and the student obliged. Our landing touchdown point was around 500 foot markers. He conducted a stabilized approach and landed normally. On our rollout; I detected no nose wheel shimmy or vibrations. We exited the runway and returned to the approach end of Runway XX for departure. After the student conducted his pre-takeoff check and made his radio call; we entered the runway and I asked him for a short field takeoff. Our departure roll appeared normal. A few seconds after rotation; I felt a bump appearing to originate from underneath the plane. On climbout; the pilot in the plane behind us advised over the radio that we may have lost a wheel. I elected to remain in the traffic pattern to further investigate the issue. During that time; an individual in a truck said over the radio that he would visually investigate to see if it was our wheel. For safety reasons; I asked the student to perform a low approach so that the individual in the truck could visually verify if we were missing one of our wheels. He stated that it appeared our nose gear was missing. I then asked the student to leave the traffic pattern and to climb to 3;000 feet. As we were more than halfway back to ZZZ1; I made the decision to return there for landing. At that time; I took the flight controls from the student. We coordinated how we would conduct the landing procedure; who would pilot the plane; and who would use the fire extinguisher and crash hammer should they be needed. We then entered the left pattern for Runway X at ZZZ1. I decided to perform one low approach so that ground and emergency personnel could verify and relay the exact condition of the nose gear. On our climbout; I [requested priority handling] over the CTAF and entered the pattern once again. I turned onto final approach. As we crossed the Runway X boundary; I had the student turn off the electric fuel pump; the electric master switch; pull the fuel shut off valve and turn off the magneto switches. I closed the throttle and pulled the mixture control. The aircraft touched down on its main wheels and I applied aft elevator pressure to hold the nose gear off the ground as long as possible. As the nose gear contacted the pavement; I was able to maintain directional control. At some point during the rollout; the nose gear fully collapsed and the aircraft came to a stop.