Let $E\in L$ be a $\Pi^1_1$ relation on $\mathbb R$. By the $\Pi^1_1$-uniformization theorem, there is $f\in L$ such that $f$ is a $\Pi^1_1$ function, $f\subseteq E$ and $dom(f)=dom(E)$ (that is, $L\models "dom(f)^L=dom(E)^L"$). By absoluteness, $V\models "f^V \text{ is a function and } f^V\subseteq E^V"$.
- Is it also true that $V\models dom(f)^V=dom(E)^V$?
- If the above is not true in general, are there any extra assumptions that will make it true?
This depend what exactly you mean by the domain are equal.
If you let $R(x,y)$ if and only if $x = y$. Then the identity function uniformizes. However the domain of this function is not the same in $L$ as it is in $V$ if $\mathbb{R}^L \neq \mathbb{R}^V$.
However in this case, the domain has a definition. It is the set of all reals. The domain of the uniformization continues to satisfy this definition in $V$ or in $L$, although they may not literally be the same set.
However, it is possible that in $L$ the domain of a uniformization has a simple definition that does not hold of the domain in $V$:
Note that relations on $\omega$ can be coded by reals. Thus you can code countable structures in the language of set theory whose domain is $\omega$.
Consider the relation $R(x,y)$ which states that $y$ is a wellfounded models of $\mathsf{ZF - P}$ and $\mathsf{V = L}$ and $x \in y$.
Note that $y$ is construed as a $\dot\in$-structure on $\omega$. $y$ has a copy of $\omega$ inside. $x$ is considered as a subset of $\omega$. $x \in y$ means there is a real in $y$ using its $\omega$ that is equal to the real $x$ on the outside.
This relation is $\Pi_1^1$ and $\mathrm{dom}(R) = \mathbb{R}^L$. In $L$, the domain is the set of all reals, but in $V$ its domain is only the set of constructible reals.
Suppose that $f$ is $\Pi_1^1$. For all $x,y \in \mathbb{R}^L$, $f(x) = y$ is $\Pi_1^1$ so by Mostowski absoluteness, $f(x) = y$ is still true in $V$. Similarly, $E(x,f(x))$ is still true in $V$ if it was true in $L$.
Note also that from the proof of $\Pi_1^1$ uniformization, given the the $\Pi_1^1$ code for the relation, there a procedure for obtain the $\Pi_1^1$ code for a function which is provably a uniformization for the the relation. So in the sense of Noah comment, the answer is yes. (I think the procedure only depends on the $\Delta_1^1$ reduction of your relation into $\mathrm{WO}$.)