I would like to recover my Gmail permanently deleted emails on iOS 15

Apple, nor backups, nor iCloud store your email messages. They are stored on the servers of your email provider. You'll need to reach out to Gmail and ask if they have a way to recover deleted messages. This isn't something Apple nor any of us users here can help you with.

If something was “permanently deleted” then it was permanently deleted.

Unless you had another copy somewhere else, it’s gone.