I don't recall either. If you're gonna hunt or fish before you get it just have proof of where you bought it. I'd think that would be ok but could double check with TWRA.
I bought mine in 2004 so don't have much to offer on how long it takes to get back. If I do recall, you are okay to hunt if you have proof that you sent it off. Call the license division in Nashville, and they can tell you for sure what you need to have on you.
I got my right before this past deer season. I called the licensing dept and asked what the turn around time was with COVID. She said scan and email the application in. They will process it and call you back for payment over the phone. It took a day or two max. Once they receive payment they update your TWRA account and it will show your lifetime license on the mobile app. I think it took a couple of weeks to get the packet from Nashville with the hard copy.
Once you got the digital copy on the app you're good to go and hunt/fish.
I sent my sons in all on paper with a check at the beginning of February and received it Two weeks ago. So took around 5 to receive it back in the mail with hard copy license.