All that "complex" gearing really isn't all that complex... its just that there's a copy for every one of the 18 cylinders. The engine was amazingly reliable in wartime use, and also in transport aircraft in the 1950s. The amazing mad genius was figuring out the combination of reciprocation and rotation that makes a sleeve valve WORK in the first place. This is an animation of the smaller Bristol Hercules, but the valves work the same way on the Centaurus:https://www.youtube.com/watch?v=_vrvep_YOio
A bit of trivia- Napier also used sleeve valves for the Sabre aircraft engine (an "H-24" configuration- basically stacked opposed 12's geared to a common output shaft). But they had terrible problems with the sleeve valves galling and seizing and the overall reliability of the Sabre was awful. Even since that era, the Bristol Centaurus has shown that it develops the same problem when air racers have tried to push more than its rated power through it, which is why the Wright 3350 is a popular re-fit for racing Sea Fury aircraft. The Wright is probably less reliable in stock form, but tolerates uprating better. Bristol tweaked everything just right for the planned rating on the Centaurus. And it sounds cool as heck.