Tower Controller reported an MSAW alert and loss of separation resulted from misunderstanding of the traffic situation.

Narrative

I told Aircraft X to continue for Aircraft Y heavy that will be holding in position. Aircraft Z that departed Runway XXR requested to come back and land Runway XY for equipment issues and was handed off to me by basic check. As I took the handoff multiple situations arose: 1. Aircraft Z was flying towards Aircraft X and was issued traffic advisory; 2. simultaneously gave Aircraft Z a low altitude alert for being low; 3. instructed Aircraft Z to do a left 360 in order to avoid Aircraft X on final and terrain to the north along with ZZZ traffic on final. Aircraft Z reported Aircraft X in sight and was instructed to follow him. I suddenly remembered that Aircraft Y heavy was holding in position as Aircraft X was continuing Runway XY and cleared him for takeoff when Aircraft X was on about a 3 mile final. Aircraft X was then cleared to land Runway XY to which he reported he was going around with no explanation at the time; and started to climb. Aircraft X was instructed to descend back down and eventually instructed to fly runway heading to maintain 3000 ft. I issued traffic for Aircraft Y heavy that departed. Aircraft X was instructed to maintain visual separation from the departed heavy. Called over to give Aircraft X 90 heading. In summary; I was caught off guard with the unusual situation and didn't have the full picture before taking the handoff for Aircraft Z. In the future; a better coordination and communication with basic check with unusual situations especially with inclement weather.