By applying suitable Liouville-type results, an appropriate parabolicity criterion, and a version of the Omori-Yau's maximum principle for the drift Laplacian, we infer the uniqueness and nonexistence of complete spacelike translating solitons of the mean curvature flow in a Lorentzian product space ${\mathbb{R}}_1\times {\mathbb{P}}_f^n$ endowed with a weight function $f$ and whose Riemannian base ${\mathbb{P}}^n$ is supposed to be complete and with nonnegative Bakry-Émery-Ricci tensor. When the ambient space is either ${\mathbb{R}}_1\times {\mathbb{G}}^n$, where ${\mathbb{G}}^n$ stands for the so-called $n$-dimensional Gaussian space (which is the Euclidean space ${\mathbb{R}}^n$ endowed with the Gaussian probability measure) or ${\mathbb{R}}_1\times {\mathbb{H}}_f^n$, where ${\mathbb{H}}^n$ denotes the standard $n$-dimensional hyperbolic space and $f$ is the square of the distance function to a fixed point of ${\mathbb{H}}^n$, we derive some interesting consequences of our uniqueness and nonexistence results. In particular, we obtain nonexistence results concerning entire spacelike translating graphs constructed over ${\mathbb{P}}^n$.