My short answer: I think not much is known. But: here is the state of the art on related problems, as far as I am aware.

Aldous and Bhamidi consider the following model. Place independent exponential edge weights on the edges of the complete graph $K_n$; we view the weights as edge lengths. Then, for each pair $u,v$ of vertices, place a constant flow between them on the shortest path from $u$ to $v$. For each edge $e$, write $F_n(e)$ for the total flow along edge $e$ in the resulting network. Then for each fixed $z > 0$, as $n \to \infty$,
```
\[
\frac{1}{n} \#\{e: F_n(e) > z\log n\} \to \int_0^{\infty}\mathbb{P}(W_1W_2e^{-u} > z)~du,
\]
```

where $W_1$ and $W_2$ are independent exponentials. (In fact, the paper proves a more detailed distributional convergence result.) In particular, *typical* edge congestion is $O(\log n)$; but the paper does not address *maximum* edge congestion. The paper also addresses vertex congestion (more directly linked to your question), showing a similar convergence result but with a more complicated term on the right-hand side of the convergence. As in the edge case, however, maximum vertex congestion is not treated, only typical congestion.

What follows is less relevant as it relates to *global* strategies for minimizing congestion, rather than greedy routing between nodes. I'm posting it anyway in case it's useful.

Alan Frieze has a survey on disjoint paths in expander graphs which may be of interest. Theorems 4 and 5 of that survey are results of Broder, Frieze and Upfal, which imply that in an expander, any set of at most $c n/\log^2 n$ pairs of vertices can be connected by disjoint paths, and that every pair can be connected by a path in such a way that the total congestion is $O(n \log n)$.

Finally, for a particular class of *random* expanders, something can be said about a fractional version (flows rather than paths; again, this allows global optimization). Given a connected, undirected graph $G=(V,E)$, a *uniform flow of volume $\phi$* on $G$ is a collection $F$ of flows, one for each ordered pair $(v,w)$ of vertices of $G$, each having volume $\phi$. Given $f \in F$ and $e \in E$, write $f(e)$ for the flow through edge $e$ in $f$ (ignore direction so this is always non-negative). Then write
```
\[
\chi(F)=\max_{e \in E} \sum_{f \in F} f(e)
\]
```

for the maximum flow across any edge of $G$, when all flows of $F$ are simultaneously active

Aldous, Mcdiarmid, and Scott have proved the following. Fix a non-negative random variable $C$ with $\mathbb{E}(C) < \infty$, and take G_n to be the complete graph $K_n$ each of whose edges $e$ is weighted with an independent copy $C_e$ of $C$. Let $\phi_n$ be the largest value such that there exists a uniform flow $F$ of volume $\phi_n$ on $G$ such that
```
\[
\sum_{f \in F} f(e) \le C_e
\]
```

for all $e \in E(K_n)$. Then there is a positive constant $\phi_*$ such that $\phi_n \to \phi_*$ in probability.

Note that if $C$ takes some fixed value $M$ with probability $p$, and is $0$ with probability $1-p$, this is equivalent to requiring maximum congestion $\le M$ on the random graph $G_{n,p}$. Thus, this setting includes (edge) congestion-type bounds on at least some expander-like graphs.