Uniformly continuous homeomorphisms

Let (X, d) be a metric space. Under which conditions is every homeomorphism from X onto X uniformly continuous with respect to (the uniformity generated by) the metric d? We give sufficient conditions for the above question and necessary conditions for it in the case of a 0-dimensional homogeneous space. It is also proved that u.c.h.-ness for every compatible metric implies compactness for a nonrigid metrizable space. Furthermore, the interplay between u.c.h.-ness and local m-compactness is considered in the class of uniform spaces.