Oh no we don't !
Quite apart from being filthy, you really don't want to be walking round cave passages for hours afterwards, giving even in v wet caves, you're only in water some of the time.
And for the OP, a wetsuit would be extremely uncomfortable to cycle in - if full lenght guarantee nasty wheals behind the knees, and anyway, you would probably die of heat stroke.
to rebut the myth that you somehow "need" to be wet for a wetsuit to work - you've missed the point. A wetsuit keeps you (somewhat) warm despite being wet, not because it's wet.
A dry wetsuit is hot, sweaty and very uncomfortable for anything resembling strenuous excercise
Regarding wind - wetsuits are basically windproof - however, a wet wetsuit, acts as a teriffic cooling device if in the wind as the evaporating water sucks heat away - hence worth wearing an anorak if (say) in a boat after diving, or if canoeing.
In summary, great lateral thinking but very bad idea though perhaps marginally better than cycling gear for deep sea diving.