There are 2 aspects here.
1. Fluid Dynamics (physic - probably not what you want).
Fluid Dynamics is a bitch. Computing how a body of water actually behaves in real-time is (mostly) beyond current hardware.
There are tricks around it - you can simulate just the surface or some sort of approximation of the volume using something like sparse voxel octrees. If you're into this sort of thing, check out good old GPU Gems (but you might want to look for something more up to date):
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2. Wet surface (graphic - probably what you do want).
If you just want to make stuff look wet - that's actually pretty easy. You just need a wet version of whatever material you are using and a mask to blend between the dry and wet look. It gets a bit messy if you need to accurately tell what part of a mesh is wet, because collisions are usually made between spheres, capsules, axis-aligned-boxes or some sort of convex hull and you need to get points in texture/uv space.
Depending on what level of realism you are going for, a wet look may be anything from bumping specular highlights and making the diffuse color darker to turning your simple metal/dielectric PBR material into a clearcoat/sss multilayered monstrosity.
It's not like this hasn't been done before, I'm sure there was a mod for Oblivion more then 10 years ago that made PC/NPC look wet if they fall into water or stand in the rain. It's just a matter of how much time and effort you want to invest in this (and how much slower your game will become). I'd be surprised if Unity and UE didn't have addons that can do all of this for you.