How do I download purchased songs from iTunes to an iPod nano?

In true Apple style, Apple Music is confusingly two separate things, although neither of them are compatible with the iPod nano:

• • Apple Music (Apple's subscription service) cannot be added to to the iPod nano. (The only iPod that can use Apple Music is the iPod Touch.) Apple Music is not "purchased", instead, the monthly subscription gives one access to Apple's large library of music. But once a user ends that subscription, they no longer have access to that music
  • Apple Music (the app, or application) is part of the suite of apps intended to replace iTunes. (Currently, it's not good enough to replace iTunes.) To use music that you have purchased (i.e from the iTunes Store, Amazon, Bandcamp etc. or from CDs) you will need to uninstall Apple Music, Apple Devices and Apple TV. Then iTunes will behave in the old way, including being able to download purchased music to the iPod nano