Interactions between light and matter allow the realization of out-of-equilibrium states in quantum solids, involving dynamic engineering of their properties. In particular, nonlinear phononics is one of the efficient approaches to realizing the stationary electronic state in non-equilibrium. Herein, by using extensive ab initio molecular dynamics, we identified that long-lived quasi-stationary geometry via light-driven nonlinear phonon dynamics could stabilize the topological nature in the material family of HgTe compounds. We show that coherent excitation of the infrared-active phonon mode results in distortions of atomic geometry with a lifetime of several picoseconds. Four Weyl points are located exactly at the Fermi level in this non-equilibrium geometry, making it an ideal long-lived metastable Weyl semimetal. We propose that such a topological phase in the light-driven state can be identified by photoelectron spectroscopy of the Fermi arc surface states or the ultrafast pump-probe transport measurements of the nonlinear Hall effect.