Many physical, chemical and biological systems have an inherent discrete spatial structure that strongly influences their dynamical behaviour. Similar remarks apply to internal or external noise, as well as to nonlocal coupling. In this paper we study the combined effect of nonlocal spatial discretization and stochastic perturbations on travelling waves in the Nagumo equation, which is a prototypical model for bistable reaction-diffusion partial differential equations (PDEs). We prove that under suitable parameter conditions, various discrete-stochastic variants of the Nagumo equation have solutions, which stay close on long time scales to the classical monotone Nagumo front with high probability if the noise level and spatial discretization are sufficiently small.