There are several statistical tests if a time series is Gaussian, although in statistics, the term "tests for normality" is usually how you search for them.
The Nist EDA site is a good place to look and the probability plot is better for shorter data sets than the sample histogram.
Near the bottom of the page, there are references to q-q plots, KS, Chi squared, and other goodness of fit tests. You can find ample information about them on the web and replicating here isn't going to add anything.
Matlab has qqplot and prob plot in the Statistics toolbox, and the qqplot with a single argument is specific to Gaussian distributions. SAS has all these tests. R has the tests.
I recommend this book, written by 2 Engineers, and they cover several tests including for things like independence, and stationarity. The book is oriented towards the practical, minimum of mathematics.
Bendat, Julius S., and Allan G. Piersol. Random data: analysis and
measurement procedures. Vol. 729. John Wiley & Sons, 2011.
The wrinkle of these tests is that they don't conform to a Signal plus Noise scenario. The tests generally assume that the time series is all Gaussian or not. A constant mean isn't a problem. Signals are not usually Gaussian and a simple test can't tell the difference.
Signal processing operations such as a DFT, tend to manifest central limit theorem effects on data, so you need to be aware that even linear transformations will not preserve a non Gaussian pdf.
It should be also noted that from a practical perspective, Gaussianity isn't black and white. Algorithms that have Gaussian assumptions usually work well even if the Gaussianity assumption is not strictly valid. Things like bi-modality and non-symmetry are more important to know about. Cauchy (heavy tails) like noise and multiplicative noise are also important to know about.