In my setup two Apple TVs and three HomePod mini were working. With the update to iOS 15.2, audioOS 15.2 and tvOS 15 one day later automations has been stopped working at all!
In the meantime I did several tests and investigation the last two weeks (since I've raised upport requests @Apple as well). I found out, as long as at least one HomePod mini is part of the setup, the issue is replicable any time. Remove the HomePod mini and remain the Apple TV as HomeKit hub, any automation will work as expected (at least a day later/assume iCloud sync over this time).
My guess: HomePod mini have problems with Thread. If a HomePod mini takes the ownership as leader w/ or w/o being a connected as HomeKit hub (you can see this in the Home app), the problem begins. As stated, remove all HomePod mini from the setup, and the problems go away after a short while.
Whether a HomePod mini or Apple TV is acting in the Thread network can be checked with the EVE-app for iOS, for instance. Unfortunately, as we all know there is no other simple option to have a look into some logs or status of Homekit.
I'm going to wait whether Apple will fix the issues with audioOS 15.3. If not I quite consider to throw them out of my setup. The situation makes me really angry. Apple support has neither provided any satisfying answer to my two service requests I've raised in the meantime more solved the problem so far.