Let $X$ be a submanifold of $M$. Inductively, let $N_0$ be the normal bundle of $X$ in $M$, and $N_{k+1}$ the normal bundle of $X$ in $N_k$. (Identify $X$ with the zero section of $N_k$, of course.)
Are the bundles $N_k$ isomorphic for all $k \in \mathbb N$? If not, are they at least isomorphic for $k$ sufficiently large?
In general, if $\pi:E\to X$ is any smooth vector bundle, then the normal bundle of the zero section is canonically isomorphic to $E$. Indeed, the tangent bundle $TE$ is canonically isomorphic to $\pi^*TX\oplus \pi^*E$ (locally this just comes from the fact that $E$ is locally $X\times\mathbb{R}^n$, and then globally the second coordinates glue together the same way local trivializations of $E$ do). Restricting to the zero section, the tangent bundle of $E$ on the zero section is canonically isomorphic to $TX\oplus E$, and so the normal bundle is canonically isomorphic to $E$.
In particular, in your setup, this gives that all the $N_k$ are canonically isomorphic.