In this paper we study the sensitivity of nonlinear stochastic differential equations of McKean-Vlasov type generated by stable-like processes. By using the method of stochastic characteristics, we transfer these equations to non-stochastic equations with random coefficients, thus making it possible to use results obtained for nonlinear PDEs of McKean-Vlasov type generated by stable-like processes in previous works. The motivation for studying sensitivity of nonlinear McKean-Vlasov SPDEs arises naturally from the analysis of the mean-field games with common noise.