No, they have to lead. In some of the worst cases there is more than a full second between pressing a key and hearing a sound. For fast music that can be several beats.
Learning an organ means you need to learn to trust the instrument (not easy when it is a mechanical beast with many parts that can fail) and feel what the music will be not react to what it is.
I suppose following a metronome instead of the drum isn't practical/possible ?