GA Flight Instructor reported initiating a go-around when observing another aircraft enter the runway when they were on short final. Instructor reported an NMAC when departing aircraft overtook them on upwind.

Narrative

While doing touch and go practice at ZZZ in a C172; a go-around was caused by traffic on the runway SR22 and a subsequent near mid-air collision. During a routine left traffic pattern for Runway XX at ZZZ; our midfield downwind call was responded to with instructions to perform a tight approach to XX for traffic. As it was a training flight; we took the opportunity to simulate a power-off 180. Upon turning final; we saw the SR22 moving towards the hold short line and eventually crossing it. Upon reviewing audio recordings; this plane was given a normal takeoff clearance for Runway XX not expedited. After we saw this aircraft on the runway; we initiated a go-around. After initiating the go-around; the Tower instructed a go-around; and we responded; stating that we were. The aircraft on the runway was not instructed to cancel their takeoff; and we were given no instruction to offset to either direction. We decided to offset slightly to the left on the upwind leg and keep our speed up as we were not sure of the location of the aircraft behind us; but we knew that the faster Cirrus would eventually overtake our C172. I was sitting on the right side and saw the aircraft pass below us and to the right; and in a bank to the right away from us; most likely an evasive action. Contributing factors. The issuance of short approach instructions to us near the same time as the takeoff clearance for the SR22; our failure to offset further to the left; the failure of ATC to issue an offset instruction; the failure of ATC to cancel the other aircraft's takeoff clearance; and the other aircraft's decision to continue the takeoff.