There is PEMCO in BHM, they are UAW, the facilty that UA uses is in Dothan, AL is also PEMCO, but I am not sure if that facility is union or not.
Back several years ago at BHM 857US an F100 was struck by lighting on landing and on landing the nose gear ripped off causing major damage to the airplane, and the UAW folks at PEMCO had a problem with US's mechanics coming down there and fixing the airplane in PEMCO's hangar being that we are IAM. They eventually made a one for one deal and got the airplane fixed.
Mobile is ST Aero, Singapore Technologies Aerospace (ST Aero), in there web page it does not mention A320 family aircraft.