There are two possible meanings of "uniform space" in the literature. I'll follow Isbell's terminology where the definition of a uniform space includes the separation axiom, and one speaks of a pre-uniform space when it is not included.
The category of pre-uniform spaces and uniformly continuous maps has concrete limits and colimits: you can take the (co)limit of the diagram of underlying sets, in the category of sets, and then endow it with the initial (final) uniform structure.
The category of uniform spaces still has concrete limits; and colimits that are generally not concrete, using a standard uniform quotient of a pre-uniform space.
Neither category is cartesian closed. The exponential law $Z^{Y\times X}\ne (Z^Y)^X$ fails in general, unless $X$ is compact. For instance, a uniformly continuous map $I\times\Bbb R\to\Bbb R$ (that is, a uniform homotopy) is not the same as a uniformly continuous map $\Bbb R\to \Bbb R^I$ (that is, a homotopy through uniformly continuous maps $\Bbb R\to\Bbb R$). Indeed, $id:\Bbb R\to\Bbb R$ is not uniformly null-homotopic, but is null-homotopic through uniformly continuous maps.
All of the above is discussed in some form in Isbell's book "Uniform spaces". For a quick review see also section 2.B in arXiv:1106.3249. Beware of the tricky nature of sequential colimits, arXiv:0908.2228.
As for complete uniform spaces, I believe they are closed under limits, but not closed under pushouts.