I’m afraid that there is no public official support document that details this. From user experience, we can deduce that in order to update automatically, your Apple TV box needs the setting for automatic updates, and also has to be on and idle for long enough to download the update files in the background without interfering with watching habits. Then the automatic update can happen within one week after update release moment/announcement (probably somewhat randomized to spread the server load for all users). Nothing will happen if the unit is in sleep mode (e.g.: week or two vacation, or when set to sleep immediately when not in active use).
Do verify manually once in a while that you have the latest updates. Update manually if needed or wanted sooner than the automatic update. (We regularly see users here that assume they have the latest updates because of the automatic setting, when it turns out after checking that latest update didn’t happen for them yet.)
About Apple TV 4K and Apple TV HD software updates - Apple Support
Update your Apple TV - Apple Support