If I understood the exotic manifolds correctly we have a following situation.
Say we have a manifold M with regular atlases $\{(U_i,\phi_i)\}_\alpha$, that are diffeomorphic to each other, and some exotic atlases $\{(\tilde{U}_i,\tilde{\phi}_i)\}_\beta$ that are diffeomorphic to each other, but not to regular atlases. Such that for $U_i\cap \tilde{U}_j$ the transformation map $\tilde{\phi}_i\circ \phi_j^{-1}$ is not a diffeomorphism.
There is a theorem, that states that a connected orientable manifold has only two orientations. Then we know that regular atlases must define two equivalence classes of orientations.
Take, for example this two definitions from Loring W. Tu: An Introduction to Manifolds:
Definition 21.9: An atlas on M is said to be oriented if for any two overlapping charts $(U,\vec{x})$ and $(V,\vec{y})$, the Jacobian determinant $det[\partial y^i/\partial x^j]$ is everywhere positive on intersection.
Definition 21.11: Two oriented atlases $\{(U_i,\phi_i)\}$, $\{(V_i,\psi_i)\}$ on a manifold M are said to be equivalent if transformation functions $ \phi_i\circ \psi_k^{-1} $ have positive Jacobian determinants for all i,k.
In the last definition we know that the transformation map and its Jacobian is well defined for regular atlases.
Now we want to know the orientation of an exotic atlas. But for the transformation map $\tilde{\phi}_i\circ \phi_j^{-1}$ the Jacobian can be undefined. This means that we have 2 equiv. classes of orientations on regular atlases, and 2 other classes on exotic atlases, and we cannot (in general) compare these pairs.
Does this mean: # of orientations = 2*(# of non-diffeomorphic smooth structures)?
For example, $S^7$ has 28 non-diffeomorphic smooth structures. Then should it have 56 orientations?
An orientation on a smooth manifold is a continuous orientation of its tangent spaces, so it depends of its smooth structure: you can't compare the orientations between your regular manifold and your exotic one, even if they have the same topology.