An air pocket exposed to water but not any other input of gasses will come to equilibrium with the proportion of gasses dissolved in water. That’s not exactly the same as the proportions of gasses in air because they don’t all dissolve equally easily (water has more CO2 for example) but it gives you an idea. I would expect an underwater bubble to eventually go away as air diffuses out a bit more easily than in.
Of course the air pockets you see in caves won’t form spontaneously underwater so the air they actually hold probably depends on where the air is coming from.
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