We study molecular outflows in a sample of 25 nearby (z < 0.17, d < 750 Mpc) ultra-luminous infrared galaxy (ULIRG) systems (38 individual nuclei) as part of the Physics of ULIRGs with MUSE and ALMA (PUMA) survey, using ∼400 pc (0.1-1.0″ beam FWHM) resolution ALMA CO(2-1) observations. We used a spectro-astrometry analysis to identify high-velocity (> 300 km s⁻¹) molecular gas disconnected from the galaxy rotation, which we attribute to outflows. In 77% of the 26 nuclei with log L_IR/L_⊙ > 11.8, we identified molecular outflows with an average v_out = 490 km s⁻¹, outflow masses 1 − 35 × 10⁷ M_⊙, mass outflow rates Ṁ_out = 6 − 300 M_⊙ yr⁻¹, mass-loading factors η = Ṁ_out/SFR = 0.1 − 1, and an average outflow mass escape fraction of 45 ± 6%. The majority of these outflows (18/20) are spatially resolved with radii of 0.2 − 0.9 kpc and have short dynamical times (t_dyn = R_out/v_out) in the range 0.5 − 2.8 Myr. The outflow detection rate is higher in nuclei dominated by starbursts (SBs, 14/15 = 93%) than in active galactic nuclei (AGN, 6/11 = 55%). Outflows perpendicular to the kinematic major axis are mainly found in interacting SBs. We also find that our sample does not follow the Ṁ_out versus AGN luminosity relation reported in previous works. In our analysis, we include a sample of nearby main-sequence galaxies (SFR = 0.3 − 17 M_⊙ yr⁻¹) with detected molecular outflows from the PHANGS-ALMA survey to increase the L_IR dynamic range. Using these two samples, we find a correlation between the outflow velocity and the star-formation rate (SFR), as traced by L_IR (v_out ∝ SFR^0.25±0.01), which is consistent with what was found for the atomic ionised and neutral phases. Using this correlation, and the relation between M_out/R_out and v_out, we conclude that these outflows are likely momentum-driven. Finally, we compare the CO outflow velocities with the ones derived from the OH 119 μm doublet. In 76% of the targets, the outflow is detected in both CO and OH, while in three targets (18%) the outflow is only detected in CO, and in one target the outflow is detected in OH but not in CO. The difference between the OH and CO outflow velocities could be due to the far-IR background source required by the OH absorption which makes these observations more dependent on the specific outflow geometry.