Context. Tracing nuclear inflows and outflows in AGNs, determining the mass of gas involved in these, and their impact on the host galaxy and nuclear black hole, requires 3-D imaging studies of both the ionized and molecular gas. Aims. We aim to map the distribution and kinematics of molecular and ionized gas in a sample of active galaxies, to quantify the nuclear inflows and outflows. Here, we analyze the nuclear kinematics of NGC 1566 via ALMA observations of the CO J:2-1 emission at 24 pc spatial and ∼2.6 km s −1 spectral resolution, and Gemini-GMOS/IFU observations of ionized gas emission lines and stellar absorption lines at similar spatial resolution, and 123 km s −1 of intrinsic spectral resolution. Methods. The morphology and kinematics of stellar, molecular (CO) and ionized ([N ii]) emission lines are compared to the expectations from rotation, outflows, and streaming inflows. Results. While both ionized and molecular gas show rotation signatures, there are significant non-circular motions in the innermost 200 pc and along spiral arms in the central kpc (CO). The nucleus shows a double-peaked CO profile (Full Width at Zero Intensity of 200 km s −1 ), and prominent (∼80 km s −1 ) blue and redshifted lobes are found along the minor axis in the inner arcseconds. Perturbations by the large-scale bar can qualitatively explain all features in the observed velocity field. We thus favour the presence of a molecular outflow in the disk with true velocities of ∼180 km s −1 in the nucleus and decelerating to 0 by ∼72 pc. The implied molecular outflow rate is 5.6 [M yr −1 ], with this gas accumulating in the nuclear 2 arms. The ionized gas kinematics support an interpretation of a similar, but more spherical, outflow in the inner 100 pc, with no signs of deceleration. There is some evidence of streaming inflows of ∼50 km s −1 along specific spiral arms, and the estimated molecular mass inflow rate, ∼ 0.1 [M yr −1 ], is significantly larger than the SMBH accretion rate (ṁ = 4.8 × 10 −5 [M yr −1 ]).