When I trigger Put to the Sword, and when does the targeted character leave play? You would react to winning the challenge by 5 or more in part D of the DUCK acronym. If it initiates successfully (no cancel), it kills that character immediately. Your opponent could interrupt that death with a "Save" effect or a Benjen/Davos/Bastard Daughter - style leave-play interrupt, but the resolution of put to the sword isn't "delayed" in any way.
D - Determine winner (and reactions to winning)
U - Claim Unopposed power (and reactions to claiming unopposed power)
C - Claim (and reactions to claim)
K - Keywords (and reactions to keywords)