There has been work done on groups whose lower uniformity (or Roelcke uniformity) is totally bounded, e.g. the orthogonal group on a Hilbert space. This condition is equivalent to saying the lower completion is compact.
In a general uniform space, locally totally bounded does not imply a locally compact completion. For example, if the upper half plane in $\mathbb C$ is given the uniformity which refines all the metrics $\|x-y\|/\|cx+d\|\|cy+d\|$ for $c,d \in \mathbb Z$ relatively prime, the completion will contain only irrational points of $\mathbb R$.
On the other hand, all the examples of groups I know which satisfy the one satisfy both. Besides locally compact groups, this includes the (affine) isometry group of a Hilbert space and the isometry group of a Hilbert hyperbolic space.
