When $N$ is the circle there's a sort of answer. In fact there's a whole chapter in the book 

Burghelea, Dan; Lashof, Richard; Rothenberg, Melvin: Groups of automorphisms of manifolds. With an appendix ("The topological category'') by E. Pedersen. Lecture Notes in Mathematics, Vol. 473. Springer-Verlag, Berlin-New York, 1975.

dedicated to showing that after looping once, the space $\text{Diff}(M\times S^1)$ splits up to homotopy as (the loops of) 
$$
\text{Diff}(M\times I) \times B\text{Diff}(M\times I) \times \eta(M) ,
$$
where the middle term is a non-connective one-fold delooping of $\text{Diff}(M\times I)$ and
$\eta(M)$ is the mysterious "nil-term" (when writing $\text{Diff}(W)$ for a manifold $W$ with boundary, the convention is that the diffeomorphisms are to preserve the boundary pointwise).
In particular, once gets a decomposition on the level of homotopy groups. (This theorem is an analog of the Bass-Heller-Swan type result which says $K(R[t]) \simeq K(R) \times BK(R) \times \eta(R)$.)


One can say something about the homotopy type the nil-term in the the concordance stable range, roughly, $\dim M/3$.


Furthermore, $\text{Diff}(M\times I)$ sits in a fibration sequence
$$
\text{Diff}(M\times I) \to C(M) \to \text{Diff}(M)
$$
where $C(M)$ is the topological group of concordances of $M$. After inverting 2, this
sequence is homotopically trivial and $\pi_k(\text{Diff}(M\times I))$ can be identified with the invariant part of the $\Bbb Z_2$-action on $\pi_k(C(M))$  induced by conjugating a concordance with the diffeomorphism which turns $M\times I$ upside down ($(x,t) \mapsto (x,1-t)$).  Lastly, $\pi_k(C(M))$ can be studied via algebraic $K$-theory methods when $k$ is within the concordance stable range.