In the case the cave is not completely flooded, you can barely tag the waterway with natural=cave or tunnel=cave anyway.
Look at this drawing from the cave here

Unfortunately the course of the river is not on this drawing, but it flows through the blue areas.
Should we go indoor micro-mapping on this, there would be a natural = cave area, a river, lakes, siphons, some paths, steps, bridges, man_made tunnels, lit=yes on some, etc…
The question remains, how to tag the river?
I’d go for waterway=underground_river or waterway=subterranean_river, news values for which the wiki definitions would emphasise the natural aspect.