Instructor pilot departing a non-towered airport reported a NMAC when another aircraft departed after and overtook them.

Narrative

My student and I were flying the RNAV 05 CGZ. On final Aircraft Y held short of Runway 05 to allow us to land. As we were on downwind I called out that we were flying the published missed approach. Aircraft Y began catching up to us according to the ADS-B on our G1000. He was quickly climbing and catching up to us. As he caught up I tried to locate him visually but could not see due to him being behind and underneath us. I asked if he was still on CTAF and received no response. I asked again if he was on CTAF and told him he was climbing towards our aircraft. I received no response so I took controls from my student and took evasive action by turning right and overflying the runway at about 4000 feet. It seemed as if he would climb right into us according to the G1000. My student was able to locate him visually at that point and we continued towards the Stanfield VOR. I do not know if he was able to see us but it seemed that he got extremely close to climbing into us according to our ADS-B information available.