Context. Tracing nuclear inflows and outflows in active galactic nuclei (AGNs), determining the mass of gas involved in them, and their impact on the host galaxy and nuclear black hole requires 3D imaging studies of both the ionized and molecular gas. Aims. We 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 favor 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 higher than the SMBH accretion rate (ṁ = 4.8 × 10⁻⁵ M_⊙ yr⁻¹).