Guide for Developer beta:
Install beta profile
From the iPhone (or iPad) paired to your HomePod, go to this page, then click on the Install Profile button below. Your iPhone will automatically switch to the Home app to install the beta profile, tap "Install" and follow the onscreen instructions.
Update your HomePod
Your HomePod will automatically receive the latest beta software if you have Automatic Updates turned on in the Software Update settings.
If you don't see the HomePod 17.5 Beta update appearing after installing the beta profile, go to the Home app > Home Settings > Software Update > HomePod Beta Updates. From there, toggle the switch to the OFF position for the HomePod speaker you want to update to HomePod17.5 Beta, then check for software updates again.
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 17.5 Public Beta or iPadOS 17.5 Public Beta.
Turn on Beta Updates
From a device running iOS or iPadOS 17.5, 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
Your HomePod 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.