There's no min/max time you can expect to wait. This has to be a very busy time of year for Tesla folks. These changing your car to put it under your account have to be the same people doing the new cars being delivered. Accessing their car through the app is part of the delivery, so I suspect they will get priority. Submit and wait. Reports from others on TMC and TM say it does not help to follow up with more emails or trying to call. That just takes them away from processing requests.
I would also make sure my driver's license image and insurance card image is uploaded under your account, in case they check to see if its already there.
If the original owner does not have another Tesla, you might talk them into giving you access to their current Tesla account letting you take it over. They need to remove their CC, driver's license and insurance card so they are not at risk. They change the password and login again, then change the email address and login again. Checking to insure they can get in and out. Then later when Tesla processes your request, you can switch back over to your own. Its been done before. Maybe the old owner will help, if they no longer have a Tesla.
Be aware, when the alias/nickname on the account is changed, the old TM forum posts will still be there but under the renamed name. No getting around that.
good luck
