Okay, so I'm not sure if you've figured out your problem yet, but I finally figured out how to fix that problem.
I also have an HTC One M8 with the same problem. When I read that you could get the music to play if you turned off "Stream Via Wifi Only" I tried that and it did the same thing.
So basically what I did was turn on "Cache during playback" and "Automatically Cache" and played each song (with my date off and wifi on) until the song progress bar (the bar that shows you how far along you are in the song, it's at the bottom of the album art when you pull up the Now Playing thing) is completely loaded. So you should be able to see it load from white to grey.
That's the easiest way I have figured out how to do it through your phone.
But quite honestly, downloading the Google Play Sync Manager onto your computer (if you have one), downloading your whole library onto said computer, and then just copying and pasting your music into any folder on your phone (which you must plug into your computer, of course) if by far the most convenient way to do it.
Hope you get/have gotten your music to work!