In this paper, nonlinear distributed joint source-channel coding (JSCC) schemes for transmission of multivariate Gaussian sources over a Gaussian multiple access channel are proposed and analyzed. The main contribution is a zero-delay JSCC named Distributed Quantizer Linear Coder (DQLC), which performs relatively close the information theoretical bounds, improves when the correlation among the sources increases, and does not level off as the signal-to-noise ratio (SNR) becomes large. Therefore it outperforms any linear solution for sufficiently large SNR. Further an extension of DQLC to an arbitrary code length named Vector Quantizer Linear Coder (VQLC) is analyzed. The VQLC closes in on the performance upper bound as the code length increases and can potentially achieve the bound for any number of independent sources. The VQLC leaves a gap to the bound whenever the sources are correlated, however. JSCC achieving the bound for arbitrary correlation has been found for the bivariate case, but that solution is significantly outperformed by the DQLC/VQLC when there is a low delay constraint. This indicates that different approaches are needed to perform close to the bounds when the code length is high and low. The VQLC/DQLC also apply for bandwidth compression of a multivariate Gaussian transmitted on point-to-point links.