We just got off the June 25 Prima cruise after an overnight in Iceland. Immigration was on the ship the day before we reached Reykjavik to clear everyone - passport stamps and all - so no other check was needed once we reached Reykjavík.
When we were ready to leave around 8am on the last day, we just tapped our card, walked off with our luggage, and boarded our transportation. Since we didn’t try to debark early, I can’t speak to if or how that would work, but they did start general debarkation for self-assist passengers around 6:30am on the last day.
As they say “your mileage may vary”. 😀