I distributed version 1.0, build 1, of my app via TestFlight, and everything worked perfectly.
Yesterday I archived and sent out build 2, and now upon tapping the update button, every user seems to be getting an error:
Unable to Download App
[App name] could not be installed at this time
I haven't heard from any users who didn't have build 1 installed already, but looking at iTunes Connect, no user has build 2 installed.
I've got this error many times, but I saw a really strange pattern and I have a pretty weird theory.
Next time, after you submit your app to Apple, don't submit it immediately for beta review. Download the app at least once with TestFlight on your iOS device as an internal tester. Do that before you submit the app for beta review and it should work.
Sounds crazy and I can't really confirm if downloading the app as an internal tester "triggers" anything, but it seems to be working for me that way. Could be a random coincidence though, would be interesting if someone can confirm/deny my theory.
Definitely one of the more annoying bugs of TestFlight right now. :/