A TRACON Controller reported an aircraft on base leg reported malfunctioning flight controls and vibrations. The aircraft turned on its own towards the airport on a conflicting heading with another aircraft on final approach.

Narrative

I was working the final sector for ZZZ. Traffic complexity and volume increased rapidly. I had all aircraft separated legally and was trying to get the final in line when Aircraft X who was on a base leg for [Runway] XXL [requested priority handling] stating that they had an issue with flight controls and a vibration. They initiated a turn towards the field and Aircraft Y who was already on a base leg. I had so much going on that I made a decision to try and run Aircraft Y across the XXL final to the XYR final to keep traffic moving. When I noticed that Aircraft Y was not descending quickly enough and that Aircraft X had increased speed I began to give rapid control instructions to Aircraft Y including a turn to the NW and an expedited descent to prevent a loss of separation all while calling the traffic to Aircraft X. Aircraft X did eventually get the traffic in sight but by that point the loss had already occurred. I do not know what the separation actually was as the radar display replay was down and I was unable to watch a replay of the event. Suggestion: The increase in traffic volume was known by people around me but as it was north operation and visual approaches nothing was done. A second final controller may or may not have changed the situation at all since it was an emergency. The sectors around me were unaware that the vast majority of my traffic was coming from the west and they chose to put Aircraft X on the western side of the final. I am not sure where Aircraft X entered the airspace but if that aircraft came in from Approach and had been fed on the eastern side to XYR I likely would not have had to write this.