Both my HomePod and HomePod Mini are showing as No Response in my Home app with no option to change that status. Does anyone have any suggestions how to get them working?

Before resetting try restarting by unplugging them. If that doesn't work try restarting your router (wifi). Wifi problems seem to be the main reason for unresponsive Homepods. There are other tips in these Discussions for troubleshooting wifi.

For example

HomePod stereo pair - one dropping out fr… - Apple Community

Sounds like they need resetting: Reset HomePod – Apple Support (AU)