TRACON Controller reported an aircraft was incorrectly conducting the published missed approach and missed a turn back to the VOR.

Narrative

I was vectoring Aircraft X for an ILS Runway XX approach at ZZZ. The pilot was planning a low approach and I assigned the published missed approach procedure as the climb-out instructions. Aircraft X checked in on the go and I assigned 7;000 ft. as their altitude assignment vice 6;600 ft. as depicted on the approach plate. I noticed Aircraft X seemed to be continuing south; past the fix ZZZZZ at which the aircraft is supposed to make a climbing right turn back to the ZZZ VOR. I asked Aircraft X if they were complying with the published missed approach procedure and they said that they were climbing to 6;600 and then beginning the turn; which is incorrect. Aircraft X appeared to be continuing further south into higher terrain and I told Aircraft X that they needed to start the turn back to the ZZZ VOR at that moment. I then observed Aircraft X begin their right turn and climb above all MVAs in the area. When Aircraft X was on a heading and at a stable altitude I advised the pilot of a possible pilot deviation and gave them the TRACON's phone number.After the incident Aircraft X wanted to go back to ZZZ1 and I do not remember if I issued a clearance back to the airport or if I assigned an altitude and a heading as an initial vector for the approach to ZZZ1. I updated Aircraft X's flight plan to show the initial approach fix for the next approach request but I may have forgotten to coordinate Aircraft X's heading in transfer to ZZZ Center's control. I should have issued a low altitude alert and told Aircraft X to immediately climb once I suspected that they were flying the missed approach procedure incorrectly. I could have asked if the pilot was in VMC conditions or able to provide their own terrain and obstruction clearance as well. It would be helpful if ZZZZZ; the fix at which aircraft need to begin the turn back to the ZZZ VOR; depicted on the radar scope so we can take action sooner if it appears a pilot is not flying the published missed approach procedure correctly.