If P = NP, does SAT, or any other NP-Complete poly-reduce to any language different from $\emptyset$ or $\Sigma^*$?
The reduction does indeed work for any non-trivial language&nbsp;$B$. And that's what the claim you've quoted says: "Let... $B$ be another language not equal to $\emptyset$ or&nbsp;$\Sigma^*$", not "... another NP language..."

So, yes, it could be arbitrarily difficult to compute $x$ and&nbsp;$y$. It might even be impossible to compute them. But that doesn't affect the running time of the reduction because the reduction doesn't have to compute $x$ and&nbsp;$y$. Rather, the person who designs the reduction has to figure out those strings once. If you think in terms of computer programs, the reduction is a program that could be arbitrarily difficult to write but, once it's been written, it runs very quickly.  In complexity theory, we don't care how difficult it is to write the program; we only care about how efficiently it runs. If the program exists, we're happy, and these programs certainly exist.

Note that it can still be possible to concretely and explictly write down the reduction, even if $B$&nbsp;is uncomputable.  For example, let $B$ be the language of descriptions of Turing machines that halt on every input. This is highly uncomputable: neither it nor its complement is recursively enumerable, for example, which means that there's no algorithm that can generate all the strings in&nbsp;$B$, or all the strings that are not in&nbsp;$B$. However, we don't need all the strings: we just need two strings.  Let $M_1$ be the Turing machine that immediately enters the halt state, regardless of its input, and let $M_2$ be the Turing machine that moves its head to the right forever, regardless of its input. We have $\langle M_1\rangle\in B$ and $\rangle M_2\rangle\notin B$ so, if you cared to figure out the values of $\langle M_1\rangle$ and $\langle M_2\rangle$ as binary strings, you could explicitly write down the reduction from an polynomial-time language to&nbsp;$B$.