There is no exact timeline, and anyone who gives you a specific number is guessing. What you are dealing with is a process that moves at the pace of the supervising agency, and that pace varies significantly depending on where you live and how backed up the caseload is.
When a registered sex offender on supervision wants to have contact with a minor, even their own child, the process is thorough by design. A probation or parole officer has to assess the situation, which typically involves a home visit, interviews, review of the circumstances of the original offense, and in some cases, input from a treatment provider or the court. Nothing gets rubber-stamped.
Weeks is a realistic minimum. Months is not unusual. The jurisdiction your father is in, the size of his supervision officer's caseload, and how quickly each step in the review process moves all affect the timeline. Some jurisdictions are faster than others, and officers with lighter caseloads can move things along more quickly.
The most important thing right now is that your father stays fully compliant with every condition of his supervision and does not push boundaries while the process plays out. Any misstep resets the clock and makes approval less likely. His cooperation and patience are the biggest factors he can actually control.
It is good that he is working on it through the right channels. That is the only way this gets resolved, and it sounds like he understands that.