Typically, about four weeks in an uncontested case and six weeks in a contested case, although this depends to some extent upon how business the relevant courts are at the time.
The time limits break down as follows into different parts of the process:
Minimum time from formally demanding possession to being legally allowed to bring suit in this situation. This is almost certainly set by statute. In Georgia, there is no waiting period for this part of the process when rent is not paid as agreed. A lawsuit can be filed the same day that a demand for possession is made by the landlord. OCGA § 44-7-50.
The time from serving a lawsuit on the tenant to the time that the tenant is required to respond in court. This is almost certainly set by statute or court rule. In Georgia this is seven days from service of process. OCGA § 44-7-51. Georgia, like most states, allows "nail and mail" service by posting a notice on the premises and mailing a notice to the last known address of the tenant, after some reasonable effort has been made to personally service the tenant or any other adult resident, so this step will typically take about seven to ten days.
The time from an appearance in court or court filing by the tenant objecting to the eviction to the time that a hearing is scheduled, in the event that the tenant raises an objection (which is easily done, even if there isn't ultimately a legally valid defense to raise). There may be a legal deadline for this in the statute or court rules, although it can usually be waived. This can also vary based on how many cases are pending relative to the number of available judges, something that usually varies seasonally. Georgia does not have a fixed time period for this step, but "Every effort should be made by the trial court to expedite a trial of the issues." OCGA § 44-7-53.
The time from an order of eviction as the conclusion of an eviction hearing until the sheriff or marshall actually carries out the eviction that has been ordered by the court. This is almost never a matter of public record or a legal mandate, and it would typically vary seasonally and over the course of a month as the staff available to carry out evictions is pretty constant, but the demand for evictions has monthly peaks that flow from month end leases and rental payment due dates, and seasonally with the end of school years and the end of calendar years being particularly busy times when delays are greater. Sheriffs have also been known to intentionally stall in holiday seasons. This part of the process in Fulton County is set forth here.
The total time is the sum of these four times.
Most evictions lawyers in the area would know what is typical for (3) and (4) and would also have a good grasp of how likely it is for a tenant to assert a defense (if the tenant does not, part (3) is skipped and a default judgment for eviction is entered on the appearance date).
In my area (in Colorado), (1) is three days, (3) is usually about two weeks and (4) is usually about one to three weeks, but as noted above, your experience in Atlanta, Georgia could easily vary based upon how busy the courts and the marshall are at that time.
According to this website, in Georgia, it is customary to allow 3-10 days for step 1 even though it is not legally required, step 3 is typically one to two weeks, and step 4 is typically about two weeks. It is also possible to extent step 2 by one week from one week to two weeks with a procedural tool that most lawyers are aware of in Georgia. So, you are typically looking at about four weeks in an uncontested case and about six in a contested case from notice to actually having people removed.