Are homeomorphic differentiable manifolds actually diffeomorphic?

Let $M$ and $N$ be two n-dimensional smooth manifolds.Suppose their underlying topological spaces are homeomorphic through $f$. Does $f$ automatically become a diffeomorphism with respect to the given smooth structures? If not, can I adjust any of the smooth structures to make $f$ a diffeomorphism? What if I restrict the manifolds to be embedded manifolds in Euclidean space $\mathbb{E}^n$ with endowed topology and smooth structure?

Let $M=N=\mathbb R$, endowed with its usual structure of a smooth manifold, and let$f:M\to N$ be the map such that $f(x)=x^3$. This is a homeomorphism but not a diffeo!

Now, if $f:M\to N$ is a homeomorphism of smooth manifolds, you can always «adjust» the smooth structures so that $f$ becomes a diffeo: indeed, you should be able to prove the following:

if $M$ is a smooth manifold, $N$ a topological space and $f:M\to N$ a homeomorphism, then there is a structure of smooth manifold on $N$ such that $f$ becomes a diffeomorphism.

In higher dimensions there exist exotic spheres which are smooth manifolds that are homeomorphic but not diffeomorphic to the $n$-sphere.

This gives a negative answer to your first question.

If by "adjust any of the smooth structure" means throw away the smooth structure on one of the manifolds and install a new unrelated one, then obviously you can use $f$ to transfer the smooth structure of one manifold to another¸ and $f$ then trivially becomes diffeomorphism. But arguably the manifold whose structure you "adjusted" wont't be the same smooth manifold anymore.

If I'm understanding things correctly, the exotic spheres can even be embedded smoothly in $\mathbb R^m$ for sufficiently large $m$.