There is no set timeline, and that uncertainty is one of the more frustrating parts of this situation for families trying to plan around it.
When an inmate completes their sentence but has an interstate detainer waiting, they do not walk out the door. They stay in custody at the current facility until the requesting state arranges transport. How long that takes depends almost entirely on logistics and resources on the other end. The requesting jurisdiction has to schedule a transport team, arrange travel, and coordinate with the holding facility. If the distance is short and the county has staff available, the pickup could happen the same day the sentence is completed. If the requesting state is across the country or is dealing with staffing constraints, it could stretch to a couple of weeks.
The holding facility is not going to release him while the detainer is active, so he stays put until transport arrives regardless of how long that takes.
One important piece of good news in this situation is that the overlap time counts. Every day he sits in the current facility after completing his original sentence, waiting for the other state to come get him, is credited as time served toward the new charge. He is not losing those days. They are accumulating in his favor and will be applied to whatever sentence or resolution comes out of the new jurisdiction.
The best thing you can do while waiting is confirm that the receiving state has been notified that he has completed his sentence and is ready for transfer. Occasionally detainer pickups get delayed simply because the paperwork notification did not move as quickly as it should have.