Light twin Instructor pilot reported a NMAC with another light aircraft on an ILS approach to AFW airport. The other aircraft reportedly was not in contact with ATC.

Narrative

There was a near midair collision while on the ILS 34R approach into AFW. There were two pilots in our aircraft. I was the instructor and the left seat pilot was also a Multiengine Instructor receiving a proficiency flight. We were on an IFR flight plan from ZZZ1 to AFW. We received vectors early on for the ILS34R and we were in the clouds for a majority of the time in the air. We were established on glideslope for the ILS 34R and cleared for the approach. As we were descending via the glideslope beyond the final approach fix we got a warning from Tower and ADSB that an aircraft was turning towards us. We were in the clouds still with no visual of the aircraft. I took controls of the aircraft applied full power and pitched up to gain altitude and separation from the aircraft at the same time that tower called and told us to maintain altitude. The aircraft appeared to be within several hundred feet vertically and horizontally on the ADSB. Initially the aircraft appeared to be flying away from the approach course but turned towards the path. The other aircraft appeared to be just below the cloud layer. Tower recleared us to land and we were still established on the glideslope and continued the approach. We broke through the clouds approximately 200ft lower and landed safely at AFW. Tower was trying to contact the aircraft as it turned towards the ILS final approach course; but did not get a response. This prompted to look closely at the ADSB traffic output. As I saw the aircraft turning towards us I prepared to level off or climb. As the aircraft continued the turn we initiated the evasive maneuver as ATC was also contacting us to stop the descent. This experience is a reminder to maintain vigilance even when you are in the IFR system and receiving traffic separation services. We had no visuals as we were in instrument meteorological conditions; but the ADSB traffic system gave us an advance warning. Moving forward I will continue to maintain vigilance in a traffic scan even if I am in instrument meteorological conditions. Crew resources were utilized effectively on this flight and I believe having a pilot flying and another pilot monitoring allowed us to avoid a more serious incident.