this paper by Howes gives a characterization of uniform spaces with a Lindelöf completion, but the characterization uses the derived uniformity, and notions like preparacompactness. I don't see as yet how it would imply, or refute, your question, but the link might help anyway. Howes did a write up of all the theory of uniform spaces and covering properties in his book "Modern Analysis and Topology" as well, and the result can also be found in its 4th chapter.
[added] It is equivalent for a Lindelöf uniform space $X$ to have a paracompact or a Lindelöf completion $X^{\ast}$, by the classical theorem (5.1.25 in Engelking, General Topology): a paracompact space that has a Lindelöf dense subspace is Lindelöf, and $X$ is dense in $X^{\ast}$.
