How can I prove that a Regular Language is closed under Union given two languages with different alphabets?
Suppose you have alphabets $\Sigma\subseteq\Sigma'$.  You can view any language $L\subseteq \Sigma^*$ as being a subset of $(\Sigma')^*$: it just happens to be language of strings that don't contain any symbols from $\Sigma'\setminus\Sigma$.

So, given languages $L_1\subseteq \Sigma_1^*$ and $L_2\subseteq\Sigma_2^*$, you can view both of these as languages over $\Sigma_1\cup\Sigma_2$, since $\Sigma_i\subseteq \Sigma_1\cup\Sigma_2$ for $i\in\{1,2\}$.  Given automata for $L_1$ and&nbsp;$L_2$, you can also modify them to work over alphabet $\Sigma_1\cup\Sigma_2$.

Now, you have two languages over the same alphabet, so you should be on familiar ground.