Nematic, or liquid-crystalline, polymer nanocomposites (NPNCs) are composed of large aspect ratio, rod-like or platelet, rigid macromolecules in a matrix or solvent, which itself may be aqueous or polymeric. NPNCs are engineered for high-performance material applications, ranging across mechanical, electrical, piezoelectric, thermal, and barrier properties. The rods or platelets possess enormous property contrasts relative to the solvent, yet the composite properties are strongly affected by the orientational distribution of the nanophase. Nematic polymer film processing flows are shear-dominated, for which orientational distributions are well known to be highly sensitive to shear rate and volume fraction of the nematogens, with unsteady response being the most expected outcome at typical low shear rates and volume fractions. The focus of this article is a determination of the ranges of anisotropy and dynamic fluctuations in effective properties arising from orientational probability distribution functions generated by steady shear of NPNC monodomains. We combine numerical databases for sheared monodomain distributions[1,2] of thin rod or platelet dispersions together with homogenization theory for low-volume-fraction spheroidal inclusions to calculate effective conductivity tensors of steady and oscillatory sheared mesophases. We then extract maximum scalar conductivity enhancement and anisotropy for each type of sheared monodomain (flow-aligned, tumbling, kayaking, and chaotic).