I rarely if ever download music from the Play Store but had a similar issue a few weeks ago trying to download a free album. After numerous failed attempts I simply gave up, it wasn't that important to me. Just wanted throw it out there that I had the same problem. Good luck guys, hopefully someone will know of an easy fix.
This seems like a weird issue with Play Music. After reading your post I realized there was some music I purchased a while back and never downloaded to my new phone. At first it would not download because of the deeply hidden "only download via WiFi" setting. Once I turned WiFi on the download started but nothing was happening. I turned WiFi off and got an error that I had no internet connection and the download stopped. Then a few seconds later I got the download successful notification. Very strange. I wonder if the issue is with the newest Play Store version? Maybe uninstall the Play Store updates? If that's even possible?
Well, you could also also try heading over to the application manager by going over to the settings> application manager> Clear the cache and the data stored by the Google PlayStore ! And see if that works out for you or not !