Meaning of $\operatorname{UOWH}(k; u_1, u_2)$
If you look up the reference [BR97] in the paper, it clearly states that $k$ is the key which is used to select a hash function from a family of hash functions and the rest of the part of the input i.e. $u_1, u_2, u_3 ...$ is the input to the selected hash such that $u_i$ is the input to $UOWH(k;u_1,u_2,...,u_i-1)$