Is there a metric space $X$ such that any $f\in C(X)$ is uniformly continuous, but there is a metric space $Y$ and $g:X\to Y$ which is continuous but not uniformly continuous?
If $X$ is compact or discrete then it never satisfies this.
Indeed, if $X$ is compact then any continuous $g:X\to Y$ is uniformly continuous. If $X$ is discrete and $\inf_{x\neq y} d(x, y)>0$ then any $g:X\to Y$ is uniformly continuous. If $\inf_{x\neq y}d(x, y) = 0$ then we can find $f:X\to\mathbb{R}$ which isn't uniformly continuous.
For further context: This is a stronger question than if two functors on uniformizable spaces, one which gives a uniformizable space its fine uniformity, and second which gives it an induced uniformity, coincide.