How to generate regular expression for language of regular expressions?
Let us consider first a simpler situation, in which you have only one "atom" $a$, only one binary operator $\circ$, and no unary operator. The expressions you are captured by
$$
X (\circ X)^*, \quad X = a+[a(\circ a)^*].
$$
In order to get an actual regular expression, substitute $X$ (surrounded by parentheses) in the left expression.

Adding a postfix operation $\star$ isn't too difficult:
$$
Y (\circ Y)^*, \quad Y = A+[A(\circ A)^*]\star^*, \quad A= a\star^*.
$$

In order to get an actual regular expression, first substitute $A$ into $Y$, then $Y$ into the leftmost expression.

Finally, the expression you are looking for only differs in that there are several "atoms" and several binary operations:
$$
Z (\circ Z)^*, \quad Z = B+[B((\cup + \circ) B)^*]\star^*, \quad B= (0+1+\underline{\epsilon}+\underline{\emptyset})\star^*.
$$
Here $\underline{\epsilon},\underline{\emptyset}$ are the versions in your alphabet.