The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it to a global thing. Anyway, I don't know how to do it.
Introduction
Ehresmann's theorem generalizes the regular value theorem. It states that there is a natural diffeomorphism $$f^{-1}(B_n) \simeq B_n \times f^{-1}(n),$$ where $n$ is a regular value of $f$, $B_n$ is its sufficiently small neighbourhood and $f$ is a smooth proper map between manifolds $f\colon M\to N$ (proper = pre-image of a compact set is compact). We can restrict to the case where $M$ is compact - then $f$ is always proper.
We can easily further generalize it to the case where we take the pre-image of (a neighbourhood of) a smooth submanifold $V$ of $N$ instead of a single point $n\in N$, assuming that the normal bundle to it is trivial: by tubular neighbourhood theorem we can find a neighbourhood of $V$ diffeomorphic to $V\times B$ (where $B$ is a ball in the normal bundle), and deduce the claim by applying the theorem to the composition $\pi_{_B} \circ f$, where $\pi_{_B}$ is the projection $V\times B\to B$.
I want to ask if this diffeomorphism may be natural in some way and if the theorem can be generalized to nontrivial bundles.
Question
To formulate the question properly, let's look at normal bundles $\def\N{\mathcal N} \N$. We can see that $$\pi_{\N(V)}\circ f_*:\N(f^{-1}(V))\to \N(V)$$ establishes a fiberwise isomorphism of the bundles (where $\pi_{\N(V)}$ is the orthogonal projection $TN\to \N(V)$) and thus we can think of $\N(f^{-1}(V))$ as a pullback of $\N(V)$. Since by the tubular nbhd thm normal bundles are diffeomorphic with the respective neighbourhoods, $f^{-1}(U)$ is some kind of pullback, but we have to go through the normal bundles and tangent maps to see this. The question is if we can represent it as a pullback naturally with respect to $f$ (not $f_*$). Formally:
Can we find such diffeomorphisms $\N(f^{-1}(V))\overset{d_M}\to f^{-1}(U)$ and $\N(V)\overset{d_N}\to U$ that the composition: $c = d_N^{-1} \circ f \circ d_M$ is "fiberwise"; i.e., it is a diffeomorphism on the fibres: $\N_{w}(f^{-1}(V)) \to \N_{f(w)}(V)$ for each $w\in f^{-1}(V)$ (not necessarily equal to $\pi_{\N(V)}\circ f_*$).
Update: I think that the orthogonal projection $\pi_{\N(V)}$ in the formula above establishing the pullback may be omitted if we choose appropriate scalar product on $TM$ (divide $T_{f^{-1}(V)}M$ into orthogonal [with respect to any scalar product] subspaces: $\mathrm{ker} f_* \oplus W$ and observe that $W$ is isomorphic to $T_{V}N$ by $f_*$ so it is enough to pull the scalar product back from $TN$ to $V$). Does it help anyhow?
Since there are requests for clarification in the comments, I will present my idea of the proof in the simple case, when the normal bundle to $V$ is trivial. The reasoning fails at one point, maybe someone could help.
By the tubular neighbourhood thm an appropriate neighbourhood of $V$ is diffeomorphic to $B\times V$, where $B$ is a ball in the normal bundle. Since $f$ is submersive, locally we can consider it as a projection and we can (locally!) arrange an $f$-related vector fields corresponding to the standard coordinate vector fields in $B$. By compactness we can assume that $B$ is small enough for these constructions. Then, via partition of unity, we can arrange global (in a neighbourhood of the whole $f^{-1}(V)$) vector fields $f$-related to the standard vector fields in $B$.
If these vector fields establish a diffeomorphism between some neighbourhood $f^{-1}(U)$ of $f^{-1}(V)$ and $B\times f^{-1}(V)$, then - in the product coordinates of respective neighbourhoods - $f$ is given by the formula $f(b,m)=(b,f(m))$.
The thing is that I don't know how to guarantee that these v.f. are "integrable" and how to treat the general case.
Edit: A possible bypass (for integrability) is to fix order of these vector fields and integrate them in that fixed order as in this proof (page 29) of Ehresmann's thm by Peter Petersen. By the IFT and compactness it should work. Am I right?