In this paper we study the impact of surface roughness in the channel on the current variability of a gate-all-around Si nanowire MOSFET. This analysis has been carried out using a fully-3D real-space non-equilibrium Green's function (NEGF) simulator. 3D simulations are required due to the strong spatial inhomogeneities of the potential and electron concentration caused by the rough interface. As an initial condition for the Green's function simulation we use a drift-diffusion solution with density gradient quantum corrections. This stabilises the convergence of the NEGF algorithm. We have obtained the ID-VG characteristics for the smooth and the surface roughness devices. A large variation in the on-current and a noticeable threshold voltage shift have been observed in the ID-VG characteristics when the smooth device is compared with the surface roughness device. The results obtained have been directly correlated with the self-consistent electrostatic potential, electron density and transmission coefficients along the wire axis.