At the time the ability triggers, determine which creature Celestial Mantle is enchanting. As the ability resolves, determine who currently controls that creature (or, if it's no longer on the battlefield, determine who controlled it when it left). That's the player whose life total is doubled. It doesn't matter who controls Celestial Mantle, who controlled the creature at the time the ability triggered, or what creature Celestial Mantle enchants at the time the ability resolves.
If a creature dealing combat damage at the same time as the enchanted creature has lifelink, the life gained due to lifelink happens before Celestial Mantle's triggered ability resolves.
If the enchanted creature's controller has a life total below 0 (which is possible if that player controls Platinum Angel, for example), Celestial Mantle's triggered ability multiplies that number by two. For example, if the player had -4 life, that player's life total becomes -8.
If a player's life total is doubled, that player actually gains or loses the necessary amount of life. For example, if the life total of the enchanted creature's controller is 14 when Celestial Mantle's triggered ability resolves, the ability causes that player to gain 14 life. Other cards that interact with life gain or life loss will interact with this effect accordingly.
In a Two-Headed Giant game, Celestial Mantle's essentially doubles the team's life total. Specifically, the triggered ability will affect one player's life total, and then the team's life total is adjusted by the amount of life the player gains as a result of this ability. Suppose a creature enchanted by Celestial Mantle deals combat damage to an opponent, and that creature's controller's team has 11 life. That player then has 11 life, so it's doubled to 22, for a net gain of 11 life. The team's life total becomes 22 (11 + 11).