I am assuming that uniform spaces are Hausdorff (although it probably doesn't matter for this question). It is more-or-less obvious that a uniform space can be embedded in a product of metric space (if d is a semi-metric on the space) form the quotient gotten by identifying pairs of points at d-distance 0 and then map into the product of these quotients), but I would be interested either to know that every uniform space can be embedded as a closed subspace of such a product or an example of one that cannot.