Semiconducting transition metal dichalcogenides can be synthesized in a wide range of structures and geometries, including closed cage nanostructures, such as nanotubes or fullerene-like nanoparticles (NPs). The latter is especially intriguing due to their stability, enhanced light-matter interactions, and ability to sustain exciton-polaritons (EPs) in ambient conditions, i.e., strong coupling of excitonic resonances to the optical cavity. Here we investigate the dynamics of EPs formation in WS2 NPs in the time domain using femtosecond transient extinction spectroscopy. We develop a gamut of analytical methods and models with time-dependent parameters to extract the underlying non-equilibrium dynamics of EPs formation. We find that the formation of EPs in WS2 NPs is not instantaneous but a gradual process that occurs only after several picoseconds. Specifically, for the short delay times, the light-matter interaction is guided by excitonic absorption, whereas for the long delay times, the process is controlled by polaritonic scattering. We discover that the coupling strength is a time-dependent entity and not a constant as is usually defined. Namely, there is a nonlinear coupling between excitonic and external modes and a notable transition from weak to strong coupling limit. Our results show that the time-dependent phenomenological dynamical model quantitatively reproduces the nonlinear dynamical coupling as well as the effects of the pump fluence on the coupling strength.