A set partition is $c$-uniform if every block has size $c$. Two families of $c$-uniform partitions of a finite set are said to be cross $t$-intersecting if two partitions from different families share at least $t$ blocks. In this paper, we establish some product-type extremal results for such cross $t$-intersecting families. Our results yield an Erdős-Ko-Rado theorem and a Hilton-Milner theorem for uniform set partitions. Additionally, cross $t$-intersecting families with the maximum sum of their sizes are also characterized.