TRACON Controller reported an aircraft cleared for an approach deviated off course below the Minimum Vectoring Altitude and approached an adjacent airport into conflict with traffic in the pattern.

Narrative

ZZZ XXR; the standard runway for this configuration; was closed due to damage. Aircraft X had been cleared for an instrument approach to Runway XXL at ZZZ. Aircraft Z was on a vector for the base leg to Runway XXL at ZZZ. Aircraft A was VFR departing ZZZ1 and talking to ZZZ1 Tower. Aircraft Y was cleared for the visual approach to Runway XYL at ZZZ2 on a right downwind talking with ZZZ2 Tower. Traffic was issued to Aircraft Z for Aircraft A as they departed ZZZ1. As Aircraft A climbed through 1000 ft. a subsequent traffic alert was issued to Aircraft Z and a turn to deconflict. Aircraft A continued to climb; while talking to ZZZ1 Tower; to 1500 ft. less than 2 miles behind Aircraft Z. As I was issuing the traffic alert to Aircraft Z I noticed that Aircraft X was no longer tracking the final approach course to ZZZ and had turned onto final for ZZZ2. This deviation placed Aircraft X in unsafe proximity to Aircraft Y. I issued a traffic alert to Aircraft X followed by a low altitude alert and an immediate climb to 2000 ft. ZZZ2 Tower informed ZZZ that Aircraft Y had Aircraft X in sight and was turned to avoid. Aircraft X climbed above the MVA; called ZZZ in sight and was cleared for the Visual Approach to XXL. Aircraft Z was issued a clearance for the ILS to XXL and he informed me that he had XXR prepared. His approach clearance was cancelled and he was re-sequenced so that he would have time to prepare and conduct a stable approach. ZZZ Tower was instructed to read the BRASHER statement to Aircraft X. ZZZ1 Tower should not depart VFR aircraft requesting RADAR services into Heavy jet aircraft. Aircraft conducting instrument approaches should not switch to a Visual Approach to a different airport without conferring with the controller.If an aircraft has listened to the ATIS broadcast announcing the closure of a runway AND hears 2 separate announcements to expect a runway other than standard due to a runway closure; should not brief the closed runway. As controllers we must continue to be vigilant; especially during other than standard operations. Multiple pilot deviations led to a cascade of events culminating in the loss of separation between multiple aircraft and an unsafe proximity to terrain.