Measurability of the pushforward operator on measures

In this case, you can say even more: under compatible Polish topologies, $\pi_*$ is not only measurable but continuous.

Indeed, suppose $U,V$ are Polish spaces and $F : U \to V$ is continuous. Then $F_* : \mathcal{P}(U) \to \mathcal{P}(V)$ is continuous in the weak topologies. The proof is immediate: suppose $\mu_n, \mu \in \mathcal{P}(U)$ with $\mu_n \to \mu$ weakly. Let $g : V \to \mathbb{R}$ be bounded and continuous. Then $g \circ F$ is a bounded continuous function on $U$, so we have $$\int_V g\,dF_* \mu_n = \int_U g \circ F \,d\mu_n \to \int_U g \circ F\,d\mu = \int_V g\,dF_*\mu.$$

For your example, fix compatible Polish topologies on $Y,Y'$, and set $U = Y \times Y'$, $V = Y$, and $F = \pi$, the projection map.