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If you accept that quantum gravity with matter should be a Topological Quantum Field Theory and that TQFTs probably can't distinguish simply connected homotopy equivalent 4-manifolds, you should come to the conclusion that at least research on quantum gravity would ultimately not depend on the smooth structures of $\mathbb{R}^4$.


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There are three facts: existence of uncountably many non-diffeomorphic exotic $\mathbf R^4$'s. any smooth manifold has a PL structure. Any PL manifold of dimension $<7$ has a smooth structure which is unique up to diffeomorphism. For the latter two facts see 1.5 and 1.8 in this survey where explicit references are given.



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