I just had this problem too and found the solution.
On the old iOS you were able to have mobile data switched off for Music but you were still able to play downloaded music. On the newer iOS you have to switch mobile data on, but don't worry you now have the option to choose exactly which kinds of music tasks you allow your mobile data to be used for.
So go to Settings > Music > Mobile Data then click the toggle so that the green shows.
When you do this you will see two more options appear below: 'Streaming' and 'Downloads' - just make sure both of those are off.
I assume Apple just needs the ability to use mobile data to verify your Apple Music membership.