The non-linearity of the electrical conductivity with applied voltage is numerically simulated for aligned multi-walled carbon nanotube (A-CNT) nanocomposites. The geometry of the reinforcement is generated based on the morphology of the A-CNT forest experimentally observed by 3D transmission electron microscope computed tomography. The polymer matrix is assumed to be electrically insulative; therefore, DC electrical conductivity is estimated by the current-voltage characteristic of the conducting CNT morphology At different values of voltage, the influence of electrical fields and magnetic fields leads to conformational changes in the nanotube network being the significant factor of the conductivity change for soft materials. The change of the tunneling resistance with the applied voltage are taken into account. The influences of the mentioned factors are comparatively analyzed. Combination of the effects provides the cumulative non-linear dependence for the nanocomposite electrical conductivity. The non-linear effects appear only for very high applied voltage, 1 V/µm and higher; the conformational change effect is felt only if the matrix on the nanocomposite is very soft, such as in foams.