General aviation Flight Instructor with student reported a near miss with another aircraft during climb out at a non-towered airport. Flight Instructor took evasive action.

Narrative

While on a practice approach with a student at Greeley; winds reported calm so we set up for RNAV 35 because the preferred calm wind runway is 35. On about a 5 mile final; traffic entered the pattern for runway 28. While on about a 2 mile final I was in contact with the traffic and informed them I have them in sight and we would initiate the missed; and shortly after began on the missed approach. I understood that the traffic was lower on final and was established in the pattern and informed them I would give way to them. Then a call was made asking when we would start the missed approach; with no tail number on the call; and I responded with 'we have started the missed and we have you in sight no factor' to which they responded with 'this is Aircraft Y (not the plane on final) we're behind you we have you in sight. I asked why they were making pattern for 28 instead of 35; someone responded with 'because that what the winds are favoring' in an aggressive tone. I knew that wasn't true because I could see the wind sock and also listened to the AWOS report twice; once on the way to the airport and once on final. The winds were calm. The traffic which was holding short of 28 began making comments directed towards my aircraft; using words like 'entitled; and student pilots' for using a different runway; and continuously spoke on the frequency; making snide remarks; inhibiting my ability to communicate with the traffic coming in behind me. Seconds later I saw Aircraft Y just above and to my left crossing into my flight path. I abruptly pushed the nose over; stopping our climb. The traffic crossed above us and we then continued the climb straight out. We discontinued the rest of the flight we had and flew back to ZZZ where we terminated the flight. I believe the traffic holding short was the biggest factor in this event because of the inability to communicate with the other traffic; along with the two aircraft talking and not using tail numbers so I was unable to discern which aircraft was talking to judge the level of threat until it was too close. Had the Aircraft Y been able to establish communication with me; or maneuver to avoid given that he called out that he had me in sight; we could have avoided the situation entirely.