We consider a joint source channel coding (JSCC) problem in which we desire to transmit an arbitrary memoryless source over an arbitrary additive channel. We propose a mismatched coding architecture that consists of Gaussian codebooks for both the source reproduction sequences and channel codewords. The natural nearest neighbor encoder and decoder, however, need to be judiciously modified to obtain the highest communication rates at finite blocklength. In particular, we consider a unequal error protection (UEP) scheme in which all sources are partitioned into disjoint power type classes. We also regularize the nearest neighbor decoder so that an appropriate measure of the size of each power type class is taken into account in the decoding strategy. For such an architecture, we derive ensemble-tight second-order and moderate deviations results. Our first-order (optimal bandwidth expansion ratio) result generalizes the seminal results by Lapidoth (1996, 1997). The dispersion of our JSCC scheme is a linear combination of the mismatched dispersions for the channel coding saddle-point problem by Scarlett, Tan and Durisi (2017) and the rate-distortion saddle-point problem by the present authors, thus also generalizing these results.