Actually, that's not true. ALPA does not have an "Aeromed department." They contract it out to a company by the name of "Aviation Medicine Advisory Service." Here's a list of their clients: AMAS Clients

I had an issue last year and worked with them briefly. What I found out was that they know their stuff, can give you good advice on your situation, can help you get the documentation together for the FAA, and can send your documentation to the FAA to go into the cue with every other aviation medical case (private pilots, corporate pilots, airline pilots, etc.) in the country. They do not appear to have any pull whatsoever in terms of "knowing guys in DC and OKC" and being able to get anything done for you. In a nutshell, all they can provide is expertise and then submit your paperwork to the FAA in OKC to wait along with everyone else.

Delta has a doctor they retain for our pilots. This doctor is just as knowledgeable (if not more so) as the AMAS doctors. And he works directly with the FAA Southern Region Flight Surgeon and can often get things looked at within days or even hours (AMAS told me it would take 3 to 4 weeks just for my file to rise to the top of the cue in OKC). I was able to get back to flying much faster by going this route and I would be very hesitant in the future to use ALPA (AMAS) when Delta already provides a much more effective resource for us.