You can get a slightly better upperbound than $\overrightarrow{\chi_{u}} (G)\leq 2^{\Delta(G)}$ by using sets of integers with distinct subset sums. For example, Bohman constructed a set $S$ of $n$ positive integers with $2^n$ distinct subset sums and with maximum element less than $0.22002\cdot2^{n}$. By taking a proper edge-colouring and assigning labels from $S$ rather than powers of $2$, we have that $\overrightarrow{\chi_{u}} (G)\leq 0.44004 \cdot 2^{\Delta(G)}$.
On the other hand, I think this is also a way to prove a super-polynomial lower bound. Define a set $S$ of positive integers to be good if for all $s \in S$, there does not exists a set $T \subseteq S \setminus \{s\}$ such that $s = \sum_{t \in T} t$. For all $n \in \mathbb{N}$, let $$g(n)=\min \{\max S : \text{$S$ is a good set of $n$ positive integers}\}.$$
Claim. For all graphs $G$, $$\overrightarrow{\chi_{u}} (G) \geq g({\Delta(G))}.$$
Proof. Let $x$ be a vertex of $G$ of maximum degree and let $S$ be the set of $\Delta(G)$ labels that appear on the edges incident to $x$. By definition, $S$ must be a good set. Therefore, $\overrightarrow{\chi_{u}} (G) \geq \max S \geq g(\Delta)$, as claimed.
I suspect that $g(n)$ is superpolynomial in $n$, although I currently do not have a proof (or reference) for this. This naturally begs the following question.
What is the growth rate of $g(n)$?