I would like to know if there exists a smooth structure on the product $\mathbb{R}^4 \times [0, 1]$ of euclidean $4$-space $\mathbb{R}^4$ and the unit interval $[0, 1]$, such that for any $s \neq t$ in $[0, 1]$ the induced smooth structures on $\mathbb{R}^4 \times \{s\}$ and $\mathbb{R}^4 \times \{t\}$ are non-diffeomorphic.
The idea is to realize at least part of the continuum of distinct smooth structures on $\mathbb{R}^4$ as a topological continuum.
Cliff Taubes established the existence of a smooth structure $\mathcal R$ on $\mathbb R^4$ and a constant $C$ so the balls $B(0,r)$ and $B(0,r')$ are pairwise non-diffeomorphic for $r,r' > C$. These are defined with respect to the usual Euclidean distance (which is continuous, but not going to be smooth on $\mathcal R$).
It seems the open subset of $(C, \infty) \times \mathcal R$ of $(t, v)$ with $d(0,v) < t$ has the form you desire.
Note that as defined the projection to the unit interval is a submersion. Note also that because $\mathbb R^5$ has a unique smooth structure, the total space is even diffeomorphic to $\mathbb R^5$.
