A TRACON Controller reported an aircraft executed a missed approach but deviated from the procedure placing it below an aircraft on approach to a nearby airport. Reporter stated they hesitated to take action due to being fatigued from working at a facility with insufficient staffing.

Narrative

I was working multiple combined positions. I heard ZZZ1 tower yell over the line that they were descending someone immediately. I observed Aircraft X climbing on the missed approach. I called local at ZZZ [tower] to alert them to the situation since I had already shipped Aircraft Y to tower. I observed a loss of separation but separation had already been reestablished. As I watched the situation unfold Aircraft X was at 1500 but didn't appear to be flying the published missed. I was unsure of what he would do or how the situation was going to be resolved since he was still flying south. I didn't want Aircraft Y to descend into him on the approach into ZZZ so I called ZZZ local back and told them to climb Aircraft Y to 4000 runway heading. There was some hesitation on both of our parts as it did eventually look like maybe they would remain separated. Knowing it was IFR and not wanting to take any risks I told local to just send him around. I was very tired after a long and exhausting shift with low staffing and busy volume all night. I believe this contributed to the hesitation; but in the end the right decision was made as both aircraft remained separated and landed safely. The Aircraft Y pilot was very appreciative on frequency when I explained what happened.It is common for go arounds at ZZZ1 to climb above the initial 1500' restriction on the missed approach. The 'top altitude' if there is such a thing on a missed approach; but there is a 1500' initial restriction to keep them separated from ZZZ arrivals. Something needs to change so that pilots aren't accidentally climbing to 3000 when they are busy with a go around as this is very dangerous with ZZZ arrivals at 3000' and descending to Runway XX. We have been briefed in the past the missed approach for arrivals to Runway XY at ZZZ1 in not procedurally separated from the ILS to Runway XXL/R at ZZZ; so we should not be running both approaches simultaneously until the procedure is changed to provide positive separation. This has been brought up in the past but the agency demands they both be used without taking any action to correct the problem. Additionally with the poor staffing in this area there is constantly too many planes to provide a safe operation; yet we are forced to just keep working too many planes hoping that nothing bad happens. The volume of traffic needs to be significantly reduced in the interest of safety and to prevent controller fatigue and burnout.