Local controller reported aircraft on go around did not respond to control instructions resulting in aircraft flying at an altitude that would be below the MVA. Aircraft finally responded to control instructions at an altitude above and turning away from the obstacle.

Narrative

I was training on Local Control. Aircraft X was on a visual approach to RWY XX.The pilot instead of joining the straight in final for RWY XX turned back to join our offset final on the RNAV Y RWY XX approach. I believe this to be the contributing factor as why the aircraft was unstable and went around. If these pilots intend to back up the visual with this approach; they should inform the radar controller so they can vector appropriately. But when they are cleared for visual expecting a straight in and then turn back across they create an unstable approach.When the aircraft went around; I instructed the trainee to ask the pilot if they would prefer to stay in the traffic pattern or to be handed off to radar for resequencing. The pilot thought for a moment and requested to stay with us in the pattern so I instructed him to enter the right downwind. I chose the right downwind because we had traffic arriving and departing RWY XY.I was discussing with my trainee what needs to be done with go arounds and also how another aircraft in our patterned needed to be handled and coordination that needed to be done with radar controller reference both aircraft.I looked up to the scope and noticed that the pilot still hadn't turned downwind as instructed and was 10 miles out on the crosswind leg. This is way further than I expected. We have heavy aircraft in the pattern all the time that remain within 5 miles. I instructed the pilot to turn downwind and told him about antennas that the aircraft was pointing at. After observing that the aircraft still hadn't turned I issued a heading to turn to avoid the antennae and an altitude of the current MVA. the aircraft was slow to take that heading and issued a stronger heading to avoid the antennae. At that time I should had also issued a new altitude of 6000 feet but I thought I was giving a heading that would turn to avoid the 6000 ft MVA. I did not issue a low altitude alert because the aircraft was at an altitude above and a heading turned away from the obstacle.Suggestion: If pilots intend to back up their visual approaches and fly the offset final instead of the straight in they should inform the radar controller of their intentions or simply just request the RNAV Y approach. This will allow for a wider and longer turn to final to help for a more stable approach or choose to use the LDA RWY XX as a back up instead.The pilot should have turned downwind sooner as instructed and the obstacle would not have been a factor. Even after I went back and instructed to turn downwind they flew a couple/few more miles before turning; again pointing them directly at the antennas. I feel that even if I had instructed the pilot to climb to 6000 ft for the next MVA it would not have changed the outcome because the aircraft never turned as instructed and was already at an altitude above the obstacle and would not have time to climb much higher by the time reaching a proximity to the obstacle.What is a reasonable pattern? 10 mile crosswind is not reasonable; and who knows how long it would have been if I didn't again instruct to turn downwind.15; 20 miles?If the pilots feel that a pattern doesn't give them sufficient time to debrief or setup then they should have requested to be resequenced with radar. (they were given the option and didn't comply)Going forward I will send Go-Arounds on our local missed approach heading and back to approach for resequencing; and train my trainees to do the same.