First off if the nail did not actually penetrate the interior of the the tire.. Just pull it out and forget it.
The nail may have twisted into the side and just punctured nothing but rubber.
If so, ZERO need to do anything but pull out the nail.
IF the nail punctured into the inner area, then it penetrated the 'ply' and that usually calls for a repair which also reduces the speed rating by one.
The sidewall area flexs a LOT and cannot be properly repaired. (Because it flexes all the time...)
I would pull out the nail now. Check the pressure a few times over the weekend, and if NO pressure is lost, the nail did NOT enter the interior of the tire.
If you are racing the car at 120mph then yeah have the tire removed and checked inside.
If all you do is normal daily driving. no problem. The tire is NOT going to come apart, explode, or otherwise cause havoc.
The TPF monitoring will tell you if it suddenly starts loosing pressure while driving.
If it does NOT lose any pressure in a few weeks, you are totally in the clear.