The law is again different from when that letter was put together. HB 1032 changed OCGA 16-11-129 to make it very clear in (d)(2) that it is the local law enforcement agency that runs a computer check (National Instant Criminal Background Check) and returns a report to the probate court. The "no report shall be required" language was left in the statute. The "Instant" part of National Instant Criminal Background Check should provide a clue as to how long this takes.
Read HB 1032 for yourself by visiting the Georgia General Assembly web site and entering HB 1032 in the upper right hand corner.