Mean geodesic distance between two points in Delaunay Graph
As you have probably come to the right conclusion by now, for a Delaunay graph, the magnitude of the mean distance between two randomly chosen nodes depends on the particular configuration of the points.

Here is one extreme example. Let each one of the six sided of the triangulated honeycomb have $n$ edges. We will have $3n(n+1)+1$ points. The average distance between two nodes is $\Theta(n)$, which is $\Theta(\sqrt {3n(n+1)+1})$.



We can also consider the row of triangles just above the middle line in that honeycomb as a separate graph. It has $4n+1$ nodes. The average distance between two nodes is $\Theta(n)=\Theta(4n+1)$.

It is intuitive clear that for any given Delaunay graph of $N$ points, the mean distance between two nodes is between $\Theta(\sqrt{N})$ and $\Theta(N)$, which grows much faster than $O(\log(N))$, the mean distance between two nodes in a small-world network. That is one evidence why we call that kind of network small world!