I was going to suggest that all the connectivity properties were either preserved or sometimes acquired by completion: e.g. a totally disconnected $X$ may become a path-connected $\bar X$, and the same is true for the other locally-defined connectivity properties I considered on that list.
But this is not true in the case of simple connectivity, or n-connectivity, because these properties depend on each point. As far as I can tell you can change them any way you like. You could put a metric on a CW-complex, but for $\bar X$ any CW complex of countably many cells, you can remove a point to change the homotopy type of $X$ as compared to $\bar X$, or just as above, let $X$ be a discrete dense set.
Or make $X$ two horizontal line segments one over the other, connected by line segments depicting an ordered bijection between dense subsets, or higher-dimensional analogues, so that $\bar X$ is a cube.
Or let $X$ be the cone of any topological space with an appropriate metric, but with the point at the tip removed, so whatever the homotopy type of $X$, $\bar X$ is contractible. I think you could even selectively remove points from a CW complex to redesign homotopy groups in more interesting ways.