Infinite union of regular language
Since every language is a countable union of regular languages, you're basically asking whether one can decide whether a given language is regular. Thus, for any reasonable representation, this task will be undecidable.

Consider the following representation of your problem, say a machine $M$ represents an infinite series of regular expressions if given input $i$, $M$ produces a regular expression $r^M_i$. You now ask whether the following language is decidable:

$L=\left\{\langle M\rangle \Big| \text{$\bigcup\limits_{i}L\left(r_i^M\right)$ is regular}\right\}$

If $M$ does not represent a series of regular expressions, i.e. on some input $i$, $M(i)$ is not a valid regular expression, then $\langle M\rangle\notin L$. $L$ is undecidable, since you can reduce the language $L_R=\{\langle M\rangle | \text{$L(M)$ is regular}\}$ to it ($L_R$ is trivially undecidable by Rice theorem). Your reduction will, given $M$, generate a machine $M'$ which on input $i$ will simulate $M$ on all inputs of length $\le i$ for $i$ steps, and according to the result, will generate a regular expression which corresponds to the union of all words that were accepted during this simulation. Clearly $\bigcup\limits_{i}L\left(r_i^{M'}\right)=L(M)$, and the validity of the reduction immediately follows.