This question here: What would the fundamental group of disjoint union look like? partly (or perhaps totally) addresses my issue, but I need anyway an explicit answer.
Problem: Let $U, V, W$ be connected Lie groups, so that, at topological level, $U$ is homeomorphic to the product $V \times W$. Let $\pi_1 (U), \pi_1 (V), \pi_1 (W)$ be their fundamental groups. Is it true that: $\pi_1 (U) = \pi_1 (V) \times \pi_1 (W)$? If yes, how is it proved, if no, what other topological restrictions on the underlying manifolds are needed to make it true?
Yes. See Hatcher Proposition 1.12.
For manifolds, connected is equivalent to path-connected, so the proposition applies.