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I'm pretty sure a lot of military submarines use inertial navigation.

I think a lot of military craft do. I know for certain a lot of jets do, and I think tanks do as well. Unsure outside of that.

It's also being used in aerospace & naval navigation. Interestingly, the more advanced systems use fibre-optic gyroscopes, see https://en.wikipedia.org/wiki/Fibre-optic_gyroscope.

Northtrop Grumman manufactures those, for example.

That seems really the only option; but how do they account for currents? Or do they match the sensor data with the engine output to correct for this?

Interesting thought experiment. I'm gonna say currents don't affect the accuracy of either a gyro or an accelerometer.