We consider weakly compressible flows coupled with a cloud system that models the dynamics of warm clouds. Our goal is to explicitly describe the evolution of uncertainties that arise due to unknown input data, such as model parameters and initial or boundary conditions. The developed stochastic Galerkin method combines the space-time approximation obtained by a suitable finite volume method with the spectral-type approximation based on the generalized polynomial chaos expansion in the stochastic space. The finite volume method takes into account the stiffness of the model due to the low Mach number flows.  To this goal, we split the flow equations into a linear stiff and a nonlinear non-stiff subsystem and apply a globally stiffly accurate IMEX time discretization. This yields an asymptotic preserving method that is uniformly stable and accurate with respect to the Mach number. The resulting numerical scheme yields a second-order accurate approximation in both space and time and exponential convergence in the stochastic space. Our numerical results demonstrate the reliability and robustness of the stochastic Galerkin method.