C172 Flight Instructor reported an ATC failure to separate traffic on a missed approach resulted in a NMAC.

Narrative

Practice approach was approved under VFR for ZZZ ILS XXR entire procedure we were flying Aircraft X for a training flight conducted under Part 91 and commenced the approach we were also cleared to fly the published missed approach procedure issued for that sector. We were handed off to ZZZ tower and after establishing comms we were 'cleared for the option XXR' the IAP was flown to DA (Decision Altitude) minimums and the missed approach procedure was commenced. At the proper altitude we began the right turn on heading after reaching appropriate altitude which is approximately midfield for our type aircraft. This put us on a 010 heading from the runway. Aircraft Y appeared to be doing practice landings and pattern work in right traffic flying at TPA (Traffic Pattern Altitude). The NMAC event happened at midfield right traffic as we were climbing out to 2000 and the other aircraft was approaching midfield in right traffic. No special instructions by ATC issued for either aircraft. Prior to NMAC ZZZ Tower ordered Aircraft X on frequency to turn left to avoid traffic conflict. Provided that we were given the full procedure and routing of the missed approach path and missed approach this traffic conflict was very likely to happen when done in conjunction with VFR traffic in right pattern for XXR or XXL at ZZZ.