My HomePod mini stuck at downloading while updating to 16.1 from 15.1.

Hello cooneys76,

Thanks for reaching out to us in Apple Support Communities about updating your HomePod. We're happy to help by looking into this more! 

Here's a resource that can help: Update HomePod - Apple Support

1. Make sure that your device is updated to the latest version of iOS, iPadOS, or macOS. 

2. Open the Home app. 

3. Tap or click the More button , then Home Settings  > Software Update.

4. If there's a new update, select Update. If there isn't a new update, you can select HomePod to turn on automatic updates and get the latest software automatically when it's available. Software Update also tells you if your HomePod is already up to date, and shows you the installed HomePod software.

A white spinning light appears on the top of HomePod while it's updating. The update might take some time. Make sure that your HomePod stays plugged in while it's updating. 

As mentioned in this article, the update can take some time, just leave it plugged in.

Hope this helps.

Best Regards.