This paper develops a Rao-Blackwellized particle filter with variational inference for jointly estimating state and time-varying parameters in non-linear state-space models (SSM) with non-Gaussian measurement noise. Depending on the availability of the conjugate prior for the unknown parameters, the joint posterior distribution of the state and unknown parameters is approximated by using an auxiliary particle filter with a probabilistic changepoint model. The distribution of the SSM parameters conditionally on each particle is then updated by using variational Bayesian inference. Experiments are first conducted on a modified nonlinear benchmark model to compare the performance of the proposed approach with other state-of-the-art approaches. Finally, in the context of GNSS multipath mitigation, the proposed approach is evaluated based on data obtained from a measurement campaign conducted in a street urban canyon.