There is no set timeline. Extradition between states moves on a bureaucratic schedule that operates quietly and unpredictably, and the inmate and their family are almost never given advance notice. Security reasons drive that secrecy. One day the person is in Missouri, and the next they are gone.
The process requires Illinois to formally request the transfer, arrange transport officers, and coordinate logistics with Missouri. That back-and-forth can take weeks or months depending on the priority level of the case and how busy both states' systems are at the time.
If there is a court date scheduled in Illinois, the transfer will almost certainly happen before that date. A judge waiting to see someone is the one thing that accelerates the timeline. Without a pending court date driving urgency, there is no legal deadline and the transfer happens when it happens.
In the meantime, the inmate is receiving credit for every day they are sitting in Missouri custody. Time is not being wasted from a sentencing standpoint, just from the perspective of everyone who is waiting for things to move.
The best way to track what is happening is through the attorney handling the Illinois case, who can check the court docket and communicate with the relevant jurisdictions.