Studying the content and distribution of molecular gas (H$_{2}$) provides key insights into how feedback from active galactic nuclei (AGNs) and star formation influences galaxy evolution, since molecular gas is the primary fuel for star formation. Ultra-luminous infrared galaxies (ULIRGs) are ideal candidates to study how AGNs and/or starbursts affect the interstellar medium due to their intense AGN and star-forming activity. We present spatially resolved multiphase molecular gas study of IRAS20551-4250, a nearby ($z=0.0429$) ULIRG, using James Webb Space Telescope (JWST)/MIRI-MRS and ALMA. Mid-infrared diagnostics do not rule out the presence of AGNs in IRAS20551-4250. [O III]$\lambda$5007 in VLT/MUSE data reveal ionized gas outflows with $w_{80}^{\rm [OIII]} \sim 790$ km s$^{-1}$ and $\dot{M}_{\rm out}^{\rm [OIII]}\lt 0.01$ M$_{\odot }$ yr$^{-1}$. No outflows are observed in either molecular phases. JWST/MIRI-MRS data reveal several rotational transitions of warm H$_{2}$ (T$\sim 500\!-\!1400$ K) within the central $\sim 4\times 4$ kpc$^{2}$ region. Excitation temperature maps suggest that the warm H$_{2}$ is primarily heated by UV radiation from the central source. The CO-based cold molecular component dominates the molecular gas mass, accounting for >95 per cent of the total molecular gas mass. Warm H$_{2}$ maps show two tidal tails and the velocity centroid maps show disturbed non-rotational motions and a systematic gradient across the field of view, similar to that of ALMA CO-based cold molecular gas and consistent with a late-stage merger. Together, our analysis indicate that the molecular gas composition in IRAS20551-4250 is consistent with ongoing star formation in the host galaxy and the outflows observed in ionized gas phase appear insufficient to expel the molecular gas or quench ongoing star formation.