Hi intercede007
I hade a closer look at the things you mentioned:
As I use Apple Music on my way to work mostly I also experienced the issues over cellular. Also, our Wi-fi in the office is not very reliable so I was using cellular most of the time as well.
I have now listened to music over wi-fi a while and unfortunately this does not seem to work either. After maybe 30 minutes the next song of an album does not play and the login screen appears after a while.
Switching back to wifi or turn on airplan mode and then switch back to wi-fi does not seem to work either.
Quite strange: even when the login prompt appread I can still listen to songs I do not own but I have been listing to recently. So Apple Music seems to cache this somehow. However: For me there is no way to play songs I did not listen to recently beside rebooting the whole device.
If this is not resolved soon I will need to quit apple music even if I like it!
I'm also on 8.4.1 but an iPhone 6.