It's pretty darn quick! (well, maybe)
My wife bought a Model Y last month from a used car dealer...so okay, maybe the car was already in some kind of "limbo" state which greased the skids a bit, but the entire process was automated. As soon as we had a signed bill of sale, we took a picture with her phone, went into her app and from her account, used the Add / Remove product to add a vehicle.
From there it pops up a dialog that says it's easiest to just have the previous owner transfer the vehicle to the new owner (remove the vehicle from their account), which is probably what you will want to do. But the "harder" method (which is what we did) was pretty darn easy. We had to upload pictures of some documents and follow the prompts in the car to add the new car, but this appeared to be completely automated, as moments later the car was in her app and off we went to the Supercharger which was only about 10 minutes away to charge up (since the car was down to about 20% SOC and we had a 50 minute drive back home).
There is a message in the app that says it can take 3-5 days, but that certainly wasn't our experience. After we followed all the instructions, it seemed instantaneous. I have to think that once the previous owner removes the car from their account, it's probably no worse than what we experienced, and it may be even easier if the previous owner actually specifically transfers the vehicle to you.
Good luck!