I have the same problem - extremely frustrating and obnoxious, by the way - and just like most if not all people with this problem, it seems to have settled in Apple Music shortly after the 14.6 update. I'm very disappointed, but I can still listen to songs I downloaded before 14.6, which is strange. The first time I noticed this bug, I had just enabled downloading songs in large ALAC files (uncompressed quality), so at first I thought that was the cause. So I disabled it - to no avail. Then my next attempt to solve the problem was to reset the audio quality of the download to AAC 256 kbit/second (standard default quality), and then re-download my music. Again, this was unsuccessful, as my music still stopped at 15 seconds. However, I had noticed another strange thing: I was saving my personal songs on my iMac, which had the same iCloud account as on my phone, which allowed me to then download and listen to my own songs in my library on my iPhone (7).After noticing the bug that stopped playing songs at 15 seconds without Wi-fi, I realized that I could listen to my personal songs at all times, while the titles were grayed out, meaning they weren't even downloaded (provided I had recently added them to my iCloud library on the iMac, otherwise they weren't all working)... while the other downloaded songs just weren't working. The situation remained the same, allowing me to listen to my own songs and songs downloaded before 14.6 without Wi-fi, but not those downloaded after the problematic update.
I imagine - and I'm not the only one with this speculation/theory - that Apple has received many complaints and grievances about this nasty bug, but that they have no solution to give, since this bug simply doesn't fix itself. All I hope is that the next update fixes this problem for good, and if so, that it arrives soon ;).