We study light scattering by cylindrical multilayer structures containing Kerr-type nonlinear materials. We develop a new semi-analytical method for solving such nonlinear problems by reducing the original 2D system by a 1D nonlinear Helmholtz equation. We apply our method for the case of wave scattering by the core-shell metal-dielectric nanowire and show that the nonlinearity allows us to control scattering cross section, which in the resonant regime demonstrates optical bistability. We compare our method with the finite-difference time-domain (FDTD) approach and find that the new approach is accurate and is 10e5 times faster and more numerically robust than the FDTD.