Equivalence for Constant-width Read-Once Branching Programs with Distinct Orders
In "Branching Programs and Binary Decision Diagrams" by Ingo Wegener [1] (very good, complete reference to check this kind of fact on branching programs), Section 5.7 deals with how you can transform a given $\pi_1$-OBDD $F_1$ into an equivalent $\pi_2$-OBDD $F_2$ by using syntactic rules. If you have a bound $B$ on the representation of $F_1$ by $\pi_2$-OBDD then you can do this in time polynomial in $|F_1|$ and $B$. He uses it to prove Corollary 5.7.11 which answers your question and a bit more: you do not need to assume the width is bounded.

Corollary 5.7.11 (in [1]): Given $F_1$ a $\pi_1$-OBDD and $F_2$ a $\pi_2$-OBDD, one can check if $F_1$ is equivalent to $F_2$ in time $O(|F_1| \cdot |F_2| \cdot \log |F_2|)$.

[1] Wegener, Ingo. Branching programs and binary decision diagrams: theory and applications. Society for Industrial and Applied Mathematics, 2000.