  We calculate the exact many-body time dynamics of polaritonic states
supported by an optical cavity filled with organic molecules. Optical,
vibrational and radiative processes are treated on an equal footing employing
the Time-Dependent Variational Matrix Product States algorithm. We demonstrate
signatures of non-Markovian vibronic dynamics and its fingerprints in the
far-field photon emission spectrum at arbitrary light-matter interaction
scales, ranging from the weak to the strong coupling regimes. We analyse both
the single and many-molecule cases, showing the crucial role played by the
collective motion of molecular nuclei and dark states in determining the
polariton dynamics and the subsequent photon emission.
