Guide for Developer beta:
Install the HomePod Sofware version 16 Profile
From the iPhone (or iPad) paired to your HomePod mini, go to BetaProfiles.com to download the configuration profile. Your iPhone will automatically switch to the Home app to install the beta profile, tap "Install" and follow the onscreen instructions.
Update your HomePod mini
Your HomePod will automatically receive the latest beta software if you have Automatic Updates turned on in the Software Update settings. To manually check for updates, tap the three-dot icon in the top right corner in the Home app from your iPhone or iPad, tap Home Settings, then go to Software Update.
You can always delete this installed profile anytime you want to stop receiving beta updates. All you need to do is go through the Delete Profile page, which has all the instructions for each of your devices.
As mentioned, Apple currently doesn't allow HomePods downgrades, so if you delete your HomePod beta profile, you will no longer receive any other beta updates until Apple officially releases them.
Provide Feedback
Once the beta is installed, it will come with a built-in Feedback Assistant app, which can be launched from the second page of your Home screen. When you come across an issue that needs addressing, send your feedback to Apple with Feedback Assistant.
Guide for Public beta:
Enroll your iOS or iPadOS device
Your iPhone or iPad will need to be running the iOS 16 public beta or iPadOS 16 public beta.
Turn on Beta Updates
From a device running iOS or iPadOS 16, go to the Home App > Home Settings > Software Update > HomePod Beta Updates and enroll the HomePod mini you want on the Beta program.
Update your HomePod mini
Your HomePod mini will automatically receive the latest public beta software if you have Automatically Update turned on in the Software Updates settings. To manually check for updates, go to Home App > Home Settings > Software Update.
Submitting Feedback
In order to submit Apple HomePod feedback, you will need the Feedback Assistant app on an enrolled iOS, iPadOS or macOS device.
After installing the iOS or iPadOS public beta, Feedback Assistant can be launched from the second page of your Home screen.
After installing the macOS public beta, Feedback Assistant can be launched from your Dock.