Cessna 172 instructor reported a NMAC while on initial climb with an aircraft on go-around.

Narrative

On Day 0 I was in a C172 conducting a training flight for a private pilot student performing touch and goes at ZZZ. The aircraft sequenced behind me (C182) was asked to perform a short approach for spacing and according to a post flight discussion with the other pilot; resulted in him not having the spacing between me on the runway and himself to perform a safe landing.The C182 then performed a go-around and advised the tower; and the tower instructed him to 'make right traffic.'As he was performing his go-around; my aircraft was already on the take off roll and upon rotation; I noticed that ADS-B was indicating another target on my position and altitude. Unclear if the other aircraft could see me or if the tower was aware of the situation; I side stepped left; and held altitude at 3-400 feet AGL until the ADS-B targets were no longer in the same position (C182 turned right crosswind); and flew a left traffic pattern to an uneventful landing.During this event; the tower controller appeared to be fixated on a Cirrus that was making confusing position calls; but was still 8-10 miles away from the airport; I believe the tower controller lost situational awareness and was not taking appropriate action to ensure that traffic could maintain visual separation. I informed the controller on frequency that we had appeared to get really close to that 182; the controller apologized. However; the C182 pilot could have proactively made a call indicating that he had the C172 in sight which would have worked to maintain see and avoid between the two aircraft.