After an immigration arrest, how soon will the undocumented immigrant be deported?

The timing and factors involved in the removal process depend on personal choices

Whether he has been ordered removed before.  ICE/immigration authorities can simply act on that previous order of removal and send him back to their country of origin immediately.

How long have they been in the United States.  If they recently came to the U.S., they might be subject to “expedited removal,” meaning that they can be deported very quickly without a hearing before an immigration judge. If, on the other hand, they have been in the U.S. for ten or more years, have been of good moral character, and can show that their deportation would cause “exceptional and extremely unusual hardship” to a qualifying relative(s) who is (or are) U.S. citizens or lawful permanent residents, they might qualify for “cancellation of removal” or a green card. Additionally, they may qualify for Temporary Protected Status (TPS) if he meets the eligibility requirements and has remained in the U.S. for a long period of time.

Whether they are a high priority for deportation.  Sometimes the U.S. government will grant prosecutorial discretion or administrative closure of cases that are not of highest priority. If the party in question has a clean criminal record and has been in the U.S. for more than three years, they might want to examine this option.

Whether they fear persecution or torture in their home country.  If so, he might avoid the effects of expedited removal, and be able to present a case for asylum, withholding of removal, or Convention Against Torture relief before an immigration judge. See  Asylum & Refugee Status  for more information on this.

Whether they would prefer to leave voluntarily, without the hassle of further proceedings.  This method avoids the effects of a removal order, such as a long-term bar to future U.S. immigration.