  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 Schr\"odinger 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.
