Let $G=(V,E)$ be a connected undirected graph such that $E$ is the union of $n$ forests $F_1\cup F_2 \cup \dots \cup F_n$. Each forest has $V$ as its nodes and containts $k$ disconnected components. Each component is simply an edge connecting 2 vertices.
is it possible to compute the chromatic polynomial of $G$ through $F_1,F_2,\dots,F_n$? since the chromatic polynomial for $F_i$ is easy to compute.
No, it already fails in very small cases and $k=1$. As an example: $K_4$ is the union of 2 spanning paths, but it is equally easy to take the union of 2 $P_4$'s and obtain a cycle, or a paw. So without detailed information about how the forests overlap, this will not be possible.
I have ignored your requirement that each component must be an edge. This would imply that $V$ must have an even number of vertices, since you also require that each forest has $V$ as its vertex set. But even if this requirement stands, the answer is still no. $K_6$ has a 1-factorization, so it is a union of five spanning forests whose components are all edges. Many other graphs can be made from five such forests, by simply letting them overlap in different ways, and these graphs do no all have the same chromatic polynomial.