How long are deleted text messages saved on the phone after they have been deleted
If you delete a message it will be deleted permanently. 

Hardware

Messages save in the memory. So once you deleted a message or any kind of file, it will be deleted permanently. But there are softwares to recover them and it is possible sometimes. The explanation of that is not suitable here since you have asked a simple question.

iCloud

If you have made a backup using iCloud and if the messages you were deleted, existed in that particular moment when that iCloud backup was made, then messages are still in the iCloud. iCloud keeps backups until you run out of iCloud storage. So you can restore those messages from iCloud anytime you want (by restoring entire backup) if those were backed up properly. If you don't need them in iCloud, you have to delete that backup.