Apple music is not compatible with google nest!
You can only connect the speaker with bluetooth and then play the music.

greetings

Hallo apple Music ist mit dem nest kompatibel.

Versuche es mal so:

On your iPhone, iPad, or Android device, open the Google Home app.
Tap Settings.
Tap Music.
Under "More music services," tap the Link icon next to Apple Music.
Tap Link Account.
Sign in with the Apple ID that you use with Apple Music and follow the onscreen steps to complete setup.