How can Wall's theorem be generalised to non-simply connected manifolds?

Gompf showed that the statement can indeed be generalized in this 1984 paper. He proved that any two smooth structures on a compact 4-manifold become diffeomorphic after taking the connected sum with some number of copies of $S^2\times S^2$.

For orientable 4-manifolds, it was shown by Gompf as Jeff says. For compact nonorientable 4-manifolds with universal cover not spin, homeomorphic also implies stably diffeomorphic.

On the other hand, Kreck showed that for every finitely presented group G with a surjective homomorphism $w \colon G \to C_2$, there is a nonorientable 4-manifold $M$ with 1-type $(G,w)$, with the following property. The 4-manifold ($M$ plus the K3 surface) is homeomorphic to ($M$ plus eleven copies of $S^2 \times S^2$), but these two manifolds do not become diffeomorphic after adding copies of $S^2 \times S^2$.

https://link.springer.com/content/pdf/10.1007/BFb0075570.pdf

Thus there is a sense in which Wall's theorem fails quite badly to generalise to nonorientable 4-manifolds.

Here 1-type means (fundamental group, orientation character).