Consider the quantity $$ \mathrm E[U \mid S,T]. $$ Is this shorthand for $$ \mathrm E[U \mid \sigma(S) \otimes \sigma(T)]? $$ If so, the defining characteristics are that $\mathrm E[U \mid S,T]$ is $\sigma(S) \otimes \sigma(T)$-measurable and $$ \mathrm E[\mathbf 1_A \, \mathrm E[U \mid S,T]] = \mathrm E[\mathbf 1_A \, U] $$ for all $A \in \sigma(S) \otimes \sigma(T)$.
Then, consider a set $B \in \sigma(T)$. Using the above definitions I don't quite see how we can write $$ \mathrm E[\mathbf 1_B \, \mathrm E[U \mid S,T]] = \mathrm E[\mathbf 1_B \, U] $$ unless $B \in \sigma(S) \otimes \sigma(T)$, which doesn't seem to make sense. This was done in this answer. The alternatives I can think of are $$ \mathrm E[U \mid \sigma(S) \cap \sigma(T)], \quad \text{or} \quad \mathrm E[U \mid \sigma(S) \cup\sigma(T)], $$ but the union isn't necessarily a sigma-algebra, and the intersection still wouldn't help unless $B \in \sigma(S) \cap \sigma(T)$, too.