There is no legal time limit. When an inmate is being held on a warrant from another county, the holding facility is not required to transfer them on any particular schedule. They can remain in custody until the next court date in the originating county is scheduled, which could be days, weeks, or longer, depending on the court's docket and how quickly the other county initiates the transfer process.
The inmate is in custody and the holding facility has no reason to rush. The other county has to formally request the transfer, arrange transport, and schedule the court appearance on their end. Until that machinery moves, nothing changes for the person sitting in the holding facility.
This waiting period is one of the more frustrating aspects of how the system handles inter-county matters. There is no number to call to speed it up and no clock the other county is required to follow. The inmate can work with their attorney to push for a quicker resolution if there is legal standing to do so, but absent that, the timeline is entirely in the other county's hands.