How soon does the pain begin when you have a kidney stone stuck somewhere?

Depends. Kidney stones can form anywhere in the urinary outflow tract. If in the pelvis of the kidney, the collection point for urine production, they can get quite large prior to any symptom occurring. Those in the ureter are usually small but big enough to partially obstruct the urine flow. These create major pain as the ureter tries to get rid of it. Stones can also form in the bladder.
May never begin. It appears that you know that you have a stone but feel no pain. If so, you may never feel pain unless the stone moves. Many stones never move.