Are all your other iCloud-based functions working? Mine were not when I migrated to the new MacBook Pro, so I had to check some boxes in the Internet Accounts>iCloud preferences. It might be worth checking, and maybe signing out and then back in. Also, I scheduled an online call with Apple Support, and it was sensationally helpful. The Apple Tech connected to my Desktop and talked me through fixing the problem. Have you considered letting Apple figure it out?
If that's not the issue, would you like the music on your hard drive for offline listening? Did you try checking the box for Automatically Download Music? What happens? Also, since you have access to your old computer, you can check to see if your old iTunes music library has files in it. They can be copied over to your new computer.