A 42-year-old woman presents to the emergency department after sustaining a traumatic injury to her left hand while operating a power saw during a home renovation project. She states she momentarily lost focus, resulting in the complete severance of her left thumb distal to the metacarpophalangeal joint. Her partner brought the detached thumb separately to the hospital. On examination, the patient is alert and oriented. Her vital signs are: blood pressure 132/84 mmHg, heart rate 98 beats/min, respiratory rate 16 breaths/min, and oxygen saturation 99% on room air. There is active, but manageable, arterial bleeding from the stump, which appears to be a clean, sharp transection. The amputated thumb appears intact with minimal contamination. Given this presentation, what is the most appropriate immediate management strategy for both the patient and the amputated part, and what critical factor significantly impacts replantation success?