We theoretically study the real-time dynamics of the photoinduced topological phase transition to a nonequilibrium Floquet Chern insulator in an organic conductor $alpha$-(BEDT-TTF)$_2$I$_3$, which was recently predicted using the Floquet theory. By using a tight-binding model of $alpha$-(BEDT-TTF)$_2$I$_3$ that hosts a pair of tilted Dirac-cone bands at the Fermi level, we solve the time-dependent Schrodinger equation and obtained time evolutions of physical quantities for continuous-wave and pulse excitations with circularly polarized light. We demonstrate that, for the continuous-wave excitations, time profiles of the Chern number and the Hall conductivity show indications of the Floquet topological insulator. We argue that the Hall conductivity exhibits a slow oscillation with its frequency corresponding to a photoinduced direct gap determined by the Floquet band structure. With pulse excitations, transient excitation spectra are obtained, from which we infer the formation of Floquet bands and the gap opening at the Dirac point during the pulse irradiation. This dynamical gap formation is also manifested by the slow oscillation component of the Hall conductivity; that is, its frequency increases with time toward the pulse peak at which it nearly coincides with the photoinduced direct gap. The relevance of the results to experiments is also discussed.
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