Jet─cloud interactions are a key manifestation of active galactic nucleus (AGN) feedback on nuclear scales, which is distinct from the large-scale radio-mode feedback that suppresses gas cooling in galaxy halos. On these smaller scales, radio jets can inject energy and momentum into the interstellar medium (ISM), shaping the physical and kinematic properties of the nuclear and circumnuclear regions of galaxies. Using JWST MIRI/MRS observations of seven nearby radio-loud AGNs (3C 293, 3C 305, Centaurus A, Cygnus A, IC 5063, NGC 1052, and M 87), we investigated jet-driven turbulence in both the warm molecular and ionized gas phases. By combining spatially resolved H2/polycyclic aromatic hydrocarbon flux ratios with diagnostic line ratios of the ionized gas, we constrained the dominant H2 excitation processes and assessed the impact of radio jet─ISM interactions on the multiphase gas. We find that radio jets drive increased turbulence in both molecular and ionized (traced by [Fe II], [Ne II], and [Ne III] lines) gas, not only along but also perpendicular to the jet axis, indicating that jet─ISM interactions extend beyond the collimated jet channel and affect the nuclear environment. Strong correlations between the H2/polycyclic aromatic hydrocarbon ratio, the H2 excitation temperature, and shock-sensitive ionized-gas tracers indicate that jet-driven shocks dominate the excitation of the H2 rotational lines in most sources. These results indicate that radio jets are a key driver of multiphase ISM kinematics and excitation in nearby radio-loud galaxies.