I had this exact same experience, but I'm especially upset because my iPhone 6s is eligible for a free battery replacement because of their manufacturing defect:
https://www.apple.com/support/iphone6s-unexpectedshutdown/
I actually set up the Genius bar appointment using a link on that web page, and I'm pretty sure that the Appointment actually got tagged with some notation about the specific issue (apparently you can't pull up past Genius bar appointment history in the Apple Store app, so I can't view that anymore).
So on the day of my appointment (Saturday, 2/10), I drive 30 minutes to the closest Apple store, get checked in, and wait another 15+ minutes for someone to get my initial info, then I wait another 10 minutes for a technician to come talk to me and tell me that they don't have any batteries in stock. He tells me that it should only take a few days to get them in and they'll contact me. In the mean time, I get an email titled "Your Apple Store Work Authorization" with the details about the battery replacement, and showing a $0 charge. A week goes by and no contact.
So yesterday (Saturday, 2/17), I tried calling the Apple Store, where I navigate their automated system and get put on hold. I'm not sure if I was really on hold with that local Apple store or some central Apple call center. After waiting on hold for over 40 minutes, I hung up.
I then tried using the Apple Support Chat feature via the Apple Store app on my iPhone. Again, I wait about 30 minutes for a Chat rep to join. I asked the Chat rep what the status of my replacement batter was and whether my local Apple store had them in stock, but all he was equipped to do was to tell me that he was going to forward my issue on to another department. Later that day I get an email which states:
Two to five weeks?!?! Per the wording of this email, it seems like I'm being stuck in the same queue as everyone who is paying for a battery replacement. Shouldn't I be given priority since my iPhone 6s is part of the replacement program (aka recall)?
Also, the repair # in this email does not match the repair # from the email I got the day of the Genius Bar appointment.