I asked this question on MSE 9 days ago and it got a very helpful comment from Eric Towers providing the Palais Stewart reference, but no answers. So I'm crossposting it here.
Let
$$
T^n \to M \to T^m
$$
be a principal torus bundle over a torus. Then $ M $ is a solvmanifold, even a nilmanifold (in fact $ M $ is the total space of a principal torus bundle over a torus if and only if it is a compact nilmanifold for a 2 step nilpotent Lie group, this is theorem 3 of [Palais, Stewart, TORUS BUNDLES OVER A TORUS]).
What if the bundle is not necessarily principal? Is every torus bundle over a torus a solvmanifold? In other words, if we have a fiber bundle
$$
T^n \to M \to T^m
$$
then can we conclude that the total space $ M $ is a solvmanifold?
EDIT:
The answer, comments and references from Igor Belegradek prove that something much stronger is true: A manifold $ M $ is the total space of a bundle
$$
N \to M \to T^n
$$
where $ N $ is a compact nilmanifold and $ T^n $ is a torus if and only if $ M $ is homeomorphic to a compact solvmanifold.
The smooth case is also addressed. Corollary 2.3 of https://arxiv.org/pdf/1307.3223.pdf states that "If $ h: T^n → T^n $ , $ n ≥ 6 $, is a diffeomorphism given by Proposition 2.1
then the mapping torus $ M_h $ is a fake torus." By fake torus the authors mean $ M_h $ is homeomorphic to a torus but not PL homeomorphic. Since $ M_h $ is not PL homeomorphic to a torus it is also not diffeomorphic to a torus. Thus already in dimension 7 we have an exotic homotopy torus $ M_h $ which is the total space of a torus bundle over the torus
$$
T^6 \to M_h \to S^1
$$
but which is not diffeomorphic to any solvmanifold. So in every dimension $ d \geq 7 $ there are smooth torus bundles over a torus (both base and fiber with the standard smooth structure) that are not diffeomorphic to any solvmanifold.
For $ d \leq 3 $ we have by Moise's theorem that there are no exotic smooth structures. So all torus bundles over a torus with total space of dimension $ d \leq 3 $ are diffeomorphic to a solvmanifold.
For $ d=4 $ the existence of exotic 4 tori (or any exotic closed aspherical manifold) seems to be an open problem see
In that same question it is pointed out that there are exotic tori in $ d=5,6 $ but it is not clear that these can be constructed as smooth fiber bundles with base and fiber standard tori.
That said, for the $ d=4 $ case a torus bundle over a torus has base and fiber with dimension $ \leq 3 $ so there shouldn't be any exotic gluing maps and it seems like the total space of such a bundle should just always have the standard smooth structure.
For $ d=5,6 $ I have no idea what kind of smooth structures might or might not be possible on torus bundles over the torus.
Best Answer
A group $G$ is isomorphic to the fundamental groups of a compact solvmanifold if and only if it fits into the short exact sequence $1\to N\to G\to\mathbb Z^n\to 1$ where $N$ is a finitely generated torsion-free nilpotent group. This is stated on p.253 and explained is chapter III of Auslander's An exposition of the structure of solvmanifolds. Part I: Algebraic theory. In particular, every torus bundle over a torus is homotopy equivalent to a solvmanifold.
You may also be interested in Theorem 3 of Wilking's paper Rigidity of group actions on solvable Lie groups which gives an analogous result for infrasolvmanifolds.