Changed Apple ID but Apple TV keeps showing old email
Probably the easiest thing to do is to just reset your Apple TV back to its factory settings.

You don't specify the exact model, so below are instructions for multiple models of Apple TV.

Apple TV 4 and Apple TV 4K


On your Apple TV go to Settings > System > Reset
You'll see two options: 


Reset: This reset your Apple TV to its factory settings and won't update the software. This can be done without an internet connection.
Reset and Update: As above, this resets your Apple TV to its factory settings, but also installs any available software updates. Obviously you will need an internet connection for this to complete.

Once your Apple TV is reset, reconfigure it by adding your new Apple ID, etc.


Apple TV models up to Apple TV 3


On your Apple TV go to Settings > General > Reset
Choose one of these options:


Reset All Settings: This option resets all settings, including your accounts.
Restore: This option resets your Apple TV to its factory settings, but also installs any available software updates. Obviously you will need an internet connection for this to complete.

Once your Apple TV is reset, reconfigure it by adding your new Apple ID, etc.