Can you compare curvatures in space, spacetime
Yes, you can compare spatial curvature with spacetime curvature. Have a look at this Baez article: "Similarly, in general relativity gravity is not really a 'force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial". There's sometimes some confusion concerning curved spacetime and curved space. Curved spacetime is not curved space and curved time. It's a curvature of "the metric". A curvature in your plot of measurements, as it were.
and hidden space?
I don't know anything about hidden space I'm afraid.
Spacetime curvature is given by the cosmological constant, that produces a De-Sitter spacetime. It is non-zero.
Spacetime curvature usually relates to the tidal force and the second derivative of gravitational potential. Have a google on that. Then check out the De Sitter universe on Wikipedia where you can read that it "models the universe as spatially flat". There's room for confusion here. I'll try to clarify...
But space curvature is nearly zero (how close to zero, compared to the cosmological constant?)
See Einstein talking about a gravitational field as space that's "neither homogeneous nor isotropic". Then see this paper which says curved spacetime is the same thing as inhomogeneous space. Then note how the FLRW metric starts with "the assumption of homogeneity and isotropy of space". That's a reasonable assumption wherein we might claim there's no overall gravitational field in the universe, because it didn't collapse when it was small and dense. Note that spacetime is curved because the universe is expanding, but that this isn't the same as the spacetime curvature of a gravitational field. In the former situation space is inhomogeneous over time, in the latter it's inhomogeneous across space.
If we accept Kaluza-Klein derived theories (I like to believe that that encompasses string theory and M-theory as well, but one is never sure these days), then one would expect that curvatures of hidden dimensions are huge to be able to stay hidden
There's no supporting evidence for string theory or M theory or any hidden dimensions.
Can we compare the curvature of a typical Kaluza-Klein theory compactification with the curvature of the cosmological constant?
I would say no, because they're chalk and cheese.
what about the curvature of the U(1) electromagnetic gauge dimension?
I would encourage you to pursue this. Check out electromagnetic geometry and things like this by Percy Hammond: "We conclude that the field describes the curvature that characterizes the electromagnetic interaction". For an analogy, imagine you're standing on a headland overlooking a flat calm sea near an estuary. You see a single wave, and you notice that its path curves a little. That's because there's a salinity gradient in the water, which is inhomogeneous. The wave is standing in for a photon, and the inhomogeneous water is standing in for a gravitational field. Now look at the surface of the sea where the wave is. It's curved.
can one express the 1/137 coupling constant of electromagnetism as a curvature?
Not directly, because it's running constant. See NIST. At an energy corresponding to the mass of the W boson : "α(mW) is approximately 1/128 compared with its zero-energy value of approximately 1/137. Thus the famous number 1/137 is not unique or especially fundamental".