Since neither iMessage nor Facetime can be activated on either device it is not possible to disable then re-enable that option or edit any settings for them, so I tried -
rebooting and resetting several times,
logging into my Apple account to check my Apple ID and password were still correct,
checking my country was correct,
changing the date manually then switching it back to Automatic,
connecting on different wi-fi networks,
disabling and re-enabling Find My Device,
switching on two-factor authentication, verifying both devices, and trying an app-specific password,
sign out and sign in again for my apple ID on the device within Settings and the Apple Store app,
downloaded and updated apps,
but nothing worked and I still can't access iMessage on my iPad mini 1 and my iPod Touch 5th Gen after updating them both to iOS9. It has been over 24 hours now.