How can $A \cup B$ be decidable if $B$ is undecidable?
If $A=\{0,1\}^*$ then $A\cup B=\{0,1\}^*$, regardless of what $B$&nbsp;is. $\{0,1\}^*$ is decidable, and choosing to express it as something involving undecidable things doesn't change that fact.

The question "Is $w$ in $A\cup B$?" is equivalent to "Is at least one of the following statements true? $w$ is in $A$; $w$ is in&nbsp;$B$; $w$ is in both $A$ and&nbsp;$B$?" In the case $A=\{0,1\}^*$, we can answer "Yes, $w$ is in&nbsp;$A$" and the rest of the questions are irrelevant.  It doesn't even matter if we can't answer "Is $w$ in&nbsp;$B$?", because we don't need to.

Here's a more practical example that might help. Suppose I give you a number&nbsp;$x$ and ask you "Is $x\in\text{Integers}\cup\text{Primes}$?" Figuring out if a large number is prime is quite difficult, but you can easily answer my question by checking if $x$&nbsp;is an integer. You can ignore the "prime" part completely.