A TRACON Controller reported a corporate jet on approach encountered icing and a loss of control. The corporate jet climbed and turned into the path of an Airbus departing an adjacent airport.

Narrative

We received a call from Flight Control Module that the BE40 was executing a go around and needed a climb out of the ice as quick as possible. At this point I observed the aircraft climbing and making a turn southwest bound with an A321 off of ZZZ southbound next them. As the BE40 checked in we observed they were continuing a turn SE bound into the path of the A321. As the pilot checked on he stated that he was having trouble maintaining control of the aircraft and needed to get out of the ice quickly. The trainee appropriately stopped the altitude and issued traffic. At this point as the trainer I took over to assist the aircraft. They leveled at 060 and were deconflicted from the A321. They were very shaken up and needed information on where they could go to stay out of the ice and ended up diverting to ZZZ1. Light to severe icing impacting the area at all altitudes between the bases at 025 and the tops 060. Suggestion: Not sure what we could have done differently in this case other than use the term traffic alert when issuing traffic. The pilot was doing their best to regain composure and control of the aircraft. The actions they took were not the instructions given to him by ATC.