Borel's conjecture predicts that anu homotopy equivalence of closed aspherical manifolds is homotopic to a homeomorphism. The conjecture has been proved for many fundamental groups, see e.g. The Borel Conjecture for hyperbolic and CAT(0)-groups by Bartels-Lueck.
Basic ingredients are topological surgery, and computations of $L$-groups and $K$-groups. Surgery works perfectly in dimensions $\ge 5$, but in dimension $4$ one is limited to fundamental groups of subexponential growth, see Subexponential groups in 4-manifold topology by Kruskal-Quinn.
I am not up to date with L-theory computations but for polycyclic groups the reference is [Farrell-Jones, The surgery L-groups of poly-(finite or cyclic) groups, Invent. Math. 91 (1988), no. 3, 559–586, EuDML].
Combining the two ingredients gives Borel's conjecture for closed 4-manifolds homotopy equivalent to an infranil 4-manifold. This includes direct products of two closed surfaces with zero Euler characteristic.
Looking at recent papers by Bartels, Farrell, and Lueck at arxiv will bring you to state of the art on the L-theory computations, in particular, see Survey on aspherical manifolds by Lueck, but I am not aware of the result that covers surface bundles specifically, and in any case one is limited to groups of subexponential growth.
