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A cartridge is reasonably water-tight (I mean, I wouldn't hide my ammo long-term at the bottom of a duck pond, but…). Smokeless powder molecules (nitrocellulose, nitroglycerine, other stuff) contain oxygen sufficient for their combustion; it does not need air to burn, only spark*.
As you can see the water gives a lot of resistance, slowing down fired bullets in short order. Moving parts such as the hammer or slide are attached to springs, and designed to operate in air. Depending on the exposed surface area moving against increased water resistance, it will either be enough to allow full cycling of the particular gun or not. Stiff springs and light re-loads would be a recipe for disaster.
*Black powder is the same, but the old expression ‘keep your powder dry’ was a reference to the non-sealed paper cartridges or open pans inhibiting the igniting spark if wet.
If memory serves on the mythbusters didn't they say the slower bullets faired a little better? The faster rounds tore themselves apart. Correct me if I'm wrong, I'm just going from memory.