do the uniformly continuous functions to the reals determine the uniformity?

It seems the answer is "Yes.", a corollary of a result in Bourbaki (Theorem IX.I.4.I):

Given a uniformity $\mathcal{U}$ on a set $X$, there is a family of pseudometrics on $X$ such that the uniformity defined by this family is identical with $\mathcal{U}$.

Now pseudometrics aren't quite functions from $X$ to $\mathbb{R}$ (they're functions from $X\times X$ to $\mathbb{R}$), but this can easily be fixed: Let $\mathcal{D}$ be a collection of pseudometrics on $X$ that induces $\mathcal{U}$. Then, $$ \left\{ d_x:d\in \mathcal{D},\ x\in X\right\} , $$ where $d_x\colon X\rightarrow \mathbb{R}$ is defined by \begin{equation} d_x(y):=d(x,y), \end{equation} is a collection of real-valued uniformly-continuous functions on $X$ from which you can recover the uniformity $\mathcal{U}$.