Your suspicion that this is a learned skill is correct -- and this even applies when performing as a soloist! The organist must learn to disregard the timing information coming to their ears, and execute playing technique all relative to what their internalized musical image (and fingers and feet) are telling them.
Also consider that (especially with antique organs), there may be quite a lot of places where latency is introduced even before the sound is actually produced. Mechanical linkages, pneumatics, all the size of a building instead of the size of a desk, like the mechanics of a piano. Modern organs do away with much of this through the use of digital technology and electric linkages where they can, but of course the speed of sound is still more or less significant depending on the size of the room.
When performing in an ensemble, problems are solved in rehearsal and by the organist's familiarity with the instrument, room, and conductor. Generally the organ pipes and choir are situated at the same end of the room, so as long as the organ and choir are playing in time, the music will be correct for all of the listeners. This may be accomplished by the conductor simply following the timing of the organ accompaniment, rather than the organist following the conductor in some cases; in others the organist may have to "guess" where to place the next note based on what happened in rehearsal.
I have never heard of an acoustic organist using a monitoring system -- keep in mind this would probably require a full organ synthesizer be involved to actually generate the monitored sound if it were happening synchronously with the keys and asynchronously from the pipes. And any pickup-based monitoring would either be late, coming from a centrally-located room mic, or completely impractical due to the sheer number of close-mic pickups you would need to effectively monitor an instrument consisting of up to tens of thousands of pipes.
The iPad item is kind of a different question, but I would consider this kind of latency to be unacceptable in digital audio for most applications. Professional audio iPad (and other computer) applications should not have noticeable latency.