We perform nonperturbative calculations of light field induced current in twisted bilayer graphene (tBLG) irradiated by a circularly polarized (CP) few-cycle laser pulse. The strong-field electron dynamics is simulated within a single-particle picture by two complementary approaches, including the velocity-gauge density-matrix equation and the length-gauge time-dependent Schr\"odinger equation, which both combine a tight-binding model for describing tBLG electronic structures. The two theoretical approaches yield the same result in the context of studying the carrier-envelope phase (CEP) and laser intensity dependent photocurrent. We show that the measured current exhibits a sinusoidal dependence on CEP of CP driving fields, with the sine phase determined by the electrode orientation with respect to the tBLG lattices. Moreover, it has been proven that an important reversal of the photocurrent direction occurs as the driving optical field strength increases to the strong-field regime, which cannot appear in the monolayer or conventional AA- and AB-stacked bilayer graphene. Based on the analysis of the conduction band population, we successfully identify that such current direction reversal in tBLG mainly originates from the interference of multichannel electron transition among two conduction bands and four valence bands closest to the Fermi level. Our results may pave the way toward investigation of nonlinear optical response in layered materials, and might provide insight for the design of future ultrafast optoelectronic devices.