If someone was incarcerated went to court and completed their time but has a hold how long before the other state comes to pick them up

There are no set time period for detainee pick-ups. Depending on the distance the sheriffs must travel to arrange for the transfer, it could be the same date of release OR it could be upto a couple of weeks before they get him. He will get time-served credit for the overlap in time towards the new charge.