This article establishes the cutoff phenomenon in the Wasserstein distance for systems of nonlinear ordinary differential equations with a unique coercive stable fixed point subject to general additive Markovian noise in the limit of small noise intensity. This result generalizes the results shown in Barrera, Högele, Pardo (EJP2021) in a more restrictive setting of Blumenthal-Getoor index $α>3/2$ to the formulation in Wasserstein distance, which allows to cover the case of general Lévy processes with some given moment. The main proof techniques are based on the close control of the errors in a version of the Hartman-Grobman theorem and the adaptation of the linear theory established in Barrera, Högele, Pardo (JSP2021). In particular, they rely on the precise asymptotics of the nonlinear flow and the nonstandard shift linearity property of the Wasserstein distance, which is established by the authors in (JSP2021). Main examples are the Fermi-Pasta-Ulam-Tsingou gradient flow and coercive nonlinear oscillators subject to small (and possibly degenerate) Brownian or arbitrary $α$-stable noise.