Suddenly cannot listen to music without wifi?

I had the exact same problem and found the answer here. Hope it's OK to do this, but I'll try to copy/paste this answer. This IS THE SOLUTION - YAY! (Of course, now I have yet ANOTHER issue w/my brand new iPod 6 128GB - still waiting for responses on that. I TOTALLY identify with your frustration!) Here goes - hope this helps . . .

Hi! I have found a solution! First, connect to wifi, and go into your iTunes app. On the bottom of the screen you will see 3 dots for "more". You will see a list of Music, TV Shows etc. Select Music, then select purchased. You will see all your music appear. On the top of the list you can select "Recent Purchases". Once you select that it will take you to the list of all the music you have purchases either on or off your phone. At the top right corner, select "Download All". Immediately your phone will start downloading each track to your phone. This will take a while so do its when you have a lot of time to kill while connected to wifi. This will enable your phone to download all your tracks at once without having to do it manually, one by one. Hope this helps, it worked for me after I got my new phone and transferred everything on my iCloud from my old phone. None of my music would play unless connected to wifi, so I did this!

Connect iPod to Mac with cable and in iPod / Summary / Options check box for "Sync only checked songs and videos." Check all the items you want and sync your iPod.