Rekindled Flame's triggered ability has an "intervening 'if' clause." That means (1) the ability won't trigger at all unless one of your opponents already has no cards in hand as your upkeep begins, and (2) the ability will do nothing if each of your opponents manages to get a card in hand by the time it resolves.

You don't specify an opponent; the ability will check all of them to see if at least one meets the condition. The opponent that meets the condition when the ability triggers and the one who meets the condition when it resolves can be different players.