GA flight instructor with student reported an NMAC during approach to MMU airport requiring the instructor to take over the controls and execute an evasive maneuver.

Narrative

While flying my instrument student on a cross country; filed IFR flight plan from ZZZ to MMU on Day 0; we had a near miss situation on final approach into MMU. We were on with ATC throughout the entire flight. Shortly prior to this incident; we were cleared by New York Approach to proceed direct to BADDA to start our approach into MMU on the ILS 23 approach. My student was completely set up for the approach as appropriate. We were switched to MMU tower. Upon contacting MMU tower; they advised us of 2 aircraft coming from our west(right side) that were descending to land at CDW. I observed both aircraft on ADS-B and observed both aircraft visually. Both aircraft were of no factor to us. Just as we were approaching the Final approach Fix (BINGG); MMU towered alerted us of an aircraft to our east (left side) climbing into the same altitude less than 1 mile away. I observed the aircraft; approximately 1/2 mile away and less than 100 feet separation in altitude. I took controls from my student; began to turn westbound; recognized that the aircraft would potentially be in the same location as I was; and decided to evade the aircraft by descending and flying below the aircraft; maintaining visual separation at all times. The other aircraft continued to climb out to the west wherever they were going. I reestablished our path on the ILS 23; stabilized the approach for the student so they could continue flying. A normal landing was made at this point; with no further issues.A note: CDW ATIS informs all pilots to 'You are advised to maintain at or below 1;500 ft. until clear of the class D airspace' since the final approach course passed 2 miles west of the airport. We were advised by MMU tower just as I began evasive maneuvers that the other aircraft had us in sight (they were advised by CDW tower) and were continuing their climb. The problem that I see is the pilot of the aircraft that came into our path did not follow the instructions of the ATIS to maintain 1500 ft. until outside of the delta; and ATC at CDW did not advise that plane not to climb either. The aircraft did not show up on our ADS-B as well. I think ATC at CDW needs to do a better job of making sure the airplanes departing their airspace when landing south need to stay at or below 1500 ft. until they have left the airspace. No aircraft should have to perform evasive maneuvers when on an instrument approach during a filed IFR flight plan.