I signed up for Apple One Premium. But when I try to use Apple Music I’m asked to subscribe to it separately. I thought Apple One covered that subscription.

I had the same issue and contacted Apple through Messages asking them for help. In walking through some troubleshooting steps, it finally worked for me but I had to go through the subscribing for Apple Music even though I had subscribed to Apple One but cancelling when it asks for confirmation (via double-clicking on the home button with Face ID authentication on my iPad Pro). The Apple Music subscription finally worked afterwards. It was a roundabout way to get my device to recognize the Apple One subscription for Music but I guess these are launch day glitches. Hope this works for you.