Outflowing shocked gas dominates the NIR H$_2$ emission from the dual AGN NGC6240
[Abridged] We present a multi-line study of the kinematics of the molecular and ionised gas phases in the central 2 kpc of NGC6240, based on JWST/NIRSpec and ALMA observations. We devised a new spectral-line fitting approach to de-blend rotating and non-rotating gas components, which is better tailored to the extreme feedback mechanisms at work in NGC6240. We find that ~65% of the Pa$\alpha$, H$_2$, and [FeII] line fluxes within the NIRSpec field of view arise from gas components that are kinematically decoupled from the stars. The NIR H$_2$ lines show the most deviation from the stars, with peak emission between the two rotating stellar structures. The PAH 3.3$\mu$m feature does not follow the NIR H$_2$ morphology, indicating that the latter does not trace PDRs. In the non-rotating gas components, we identify a biconical wind launched from the northern AGN, expanding along the minor axis of stellar rotation. This wind is dominated by ionised gas and, although it entrains some H$_2$, it does not show a H$_2$/PAH enhancement, suggesting either high UV irradiation or expansion along a relatively gas-free path. Furthermore, we find bright non-rotating gas emission between the two AGN and around the southern AGN, which we interpret as due to an outflow launched from the southern nucleus, coinciding with the molecular outflow previously studied in cold (sub-)millimeter tracers. The strong H$_2$/PAH enhancement measured in this region, coextensive with high velocity redshifted gas ($v\sim900$ km s$^{-1}$), suggests that the shocks responsible for the high H$_2$/PAH ratios are due to the outflow rather than to the collision of media during the merger. Our results show that the bulk of the NIR line emission in NGC6240 is decoupled from the stars, and that most of the warm H$_2$ is shock-excited and embedded in a powerful outflow, where it coexists with colder molecular gas.