Is the atmospheric voltage gradients beneath overcast sky higher or lower than it is on a clear day? According to Wikipedia and other sources the atmospheric electric voltage gradient near the Earth's surface is approx 100 V/m on a 'fair weather' sort of day. Does a diffusely overcast and cloudy sky make that value higher or lower?
 A: A diffusely overcast or cloudy sky will typically lower the atmospheric vertical electric field near the earth's surface.
There are lots of variations in the electric field that depend on the weather, but you specifically ask about diffuse overcast or cloudy sky and don't mention any precipitation, so let's consider the case of simple uniform clouds.
Local atmospheric vertical electric fields and currents are part of a global atmospheric electric circuit in which the "batteries" are  electrified nimbostratus clouds (e.g. thunderstorms) that maintain an average ionospheric electric potential of about 240 KV which drives a fair-weather current flow of order $10^{-12}\textrm{A/m}^2$.  The atmosphere is essentially a leaky capacitor with the top and bottom conductive plates being the ionosphere and the earth's surface. The air in between is weakly conductive because of ions that are produced by cosmic rays and terrestrial radiation.  Inside clouds, these ions attach themselves to cloud droplets which are larger and less mobile. This increases the resistivity of the atmosphere within the cloud, so stratus clouds  act as vertical electrical resistors.
As in any DC circuit, the relative voltage drop across any circuit component is proportional to its resistance, so the vertical electric field within a simple cloud will increase compared to the atmosphere outside the cloud.
Fog is essentially a cloud sitting on the Earth's surface, and the vertical electric field within fog can be as much as 1000 V/m.
On the other hand, when there are clouds overhead, you have basically added a large resistance to the vertical atmosphere above you which means that more of the total atmospheric potential will drop across them and less will drop near the surface. This reduces the electric field near the surface by up to 30% or more.
This simple model will start to fail, however, if there is any local precipitation, thunderstorms, active cloud formation, pollution, volcanos, wind, ocean spray, ….
