We report on experimental and theoretical studies of the fluctuation-induced escape time from a metastable state of a nanomechanical Duffing resonator in a cryogenic environment. By tuning in situ the nonlinear coefficient γ we could explore a wide range of the parameter space around the bifurcation point, where the metastable state becomes unstable. We measured in a relaxation process the distribution of the escape times. We have been able to verify its exponential distribution and extract the escape rate Γ. We investigated the scaling of Γ with respect to the distance to the bifurcation point and γ, finding an unprecedented quantitative agreement with the theoretical description of the stochastic problem. Simple power scaling laws turn out to hold in a large region of the parameter space, as anticipated by recent theoretical predictions. These unique findings, implemented in a model dynamical system, are relevant to all systems experiencing underdamped saddle-node bifurcation.