Polaron formation in pump-probe experiments is an inherently non-equilibrium phenomenon, driven by the ultrafast coupled dynamics of electrons and phonons, and culminating in the emergence of a localized quasiparticle state. In this work, we present a first-principles quantum-kinetic theory of polaron formation that captures the real-time evolution of electronic and lattice degrees of freedom in presence of electron-phonon coupling. We implement this framework to investigate the ultrafast polaron formation in the prototypical polar insulator MgO. This approach allows us to determine the characteristic timescales of polaron localization and to identify its distinctive dynamical fingerprint. Our results establish clear and experimentally accessible criteria for identifying polaron formation in pump-probe experiments.