The redeem procedure doesn’t depend on any email. It is just a reminder, to redeem within the redeem time window (first three months).
To get the offer, redeem on your new eligible Apple device, not necessarily the device where you may have received the email message.
The duration of the offered free period is different (shorter) for devices activated on or after July 1, 2021.
Read here how to redeem in more detail: Apple TV+ offer - Apple
Read here about the conditions: Apple TV+ offer - Terms and conditions - Apple
Read here if you have trouble redeeming: If you can't redeem your promotion for an Apple subscription - Apple Support