When lattices of uniformly continuous functions on X determine X

Several Banach-Stone-like generalizations of Shirota's result for metrizable uniform spaces are proved. Namely, if complete uniform spaces X,Y have isomorphic lattices U(X),U(Y) of their real-valued uniformly continuous functions, and both X,Y are either some products of spaces having monotone bases (metrizable or uniformly zero-dimensional), or are locally fine and of non-measurable cardinality, then X and Y are uniformly homeomorphic.