Finding nd - m dimensional neighbors for a given node within a balanced hyperoctree
You are asking two different questions:


What are the $2^{n_d} \binom{n_d}{m}$-many codimension $m$ neighbors of a given node?
Given a description of a neighbor, how to locate it?


Regarding the first question, a neighbor of codimension $m$ is given by (1) a subset $S$ of the coordinates of size $m$, (2) for each $x \in S$, a direction $d_x \in \{-1,+1\}$. The address of a neighbor is obtined by adding $d_x$ to coordinate $x$.

Regarding the second question, the simplest algorithm would consist of the following steps:


Find the address of the node by traversing the tree up.
Find the address of the neighbor. This part is described above.
Find the neighbor by traversing the tree down.


These steps are straightforward generalizations of the algorithm described in the link.