HOW LONG DOES THE PROCESS TAKE IF I TAKE MY SON'S FATHER TO COURT TO SIGN PASSPORT PAPERS

You will have to file a petition in court to request a court order to compel the father to sign the passport papers. A motion may take a month or more to be heard, however, you can attempt to file it as an emergency application by showing irreparable harm if the court does not hear the case sooner. This would best be done with an attorney. You will likely be required, if approval is granted, to give the child's father access to your itinerary, an opportunity to keep up contact with his son, especially if visits or contacts are being cut off during the time, etc. If there are places you are planning to go to that might be dangerous, the court may have concerns as well.