This work presents a general methodology to analyze three-dimensional Freedericksz transitions in twisted-nematic liquid crystal (LC) bilayers. Using two equivalent coupled electromechanical variational formulations, the problem is treated as a bifurcation instability triggered by an externally applied electric field. Specifically, we consider LC bilayer materials anchored between two bounding plates and subjected to an electric field across the bilayer thickness. The plates are also twisted by an overall angle leading to different orientations of the directors in each layer. We first evaluate the corresponding ground state of the director field, and subsequently, we analyze the bifurcation problem by using a combined analytical-numerical method leading to a one-dimensional finite element discretization of the resulting stiffness matrix of the system. An analytical solution for the zero-twist bilayer is also obtained. The developed methodology is used to study the effect of the volume fraction of the constituents forming the bilayer upon the resulting critical electric field and corresponding eigenmodes. We find that by assembling a relatively thin 5CB layer with a thicker 7E layer, one can obtain periodic Freedericksz transitions (i.e. local modes) even for a zero-twist LC bilayer. We also show that when a 5CB material is assembled together with another electrically similar LC, such as a PCH12, the combined LC bilayer can exhibit an even lower Freedericksz transition than a LC of the same thickness consisting of any of the two constituents alone.