Military Helicopter pilot reported an NMAC during approach to a non-towered airport. A higher performance aircraft on the same approach caused the conflict requiring the reporter to take evasive action.

Narrative

I was instructing in a military training helicopter cleared for the LOC XX into ZZZ. No ceiling and visibility was unrestricted. Shortly after being cleared but before being pushed over to CTAF by Approach; I noticed an aircraft coming up behind us on the same course and altitude at a faster rate of speed on our traffic page. I told my student to descend and while doing so I swapped to the CTAF frequency on our second radio and announced something along the lines of; 'ZZZ traffic; to the aircraft on a 7 miles final at about 2000 feet; this is the aircraft on a 6.5 mile final now about 100 feet below you'. They responded saying they were shooting the ILS XX inbound and flew over top of us within 200-300 feet in a small fixed wing aircraft. I missed the rest of their call as approach told us to swap over to advisory at that point. Using my display I recorded what I believe to be the tail number of the other aircraft; finished our approach and returned home. The other aircraft landed at ZZZ. We received no warning from Approach about the closing traffic despite being on an IFR flight plan. Had I not noticed the approaching aircraft on my display and told my student to descend; we would have had a collision or come very close to a collision.