Uniform semi-continuity

A little further thought reveals the following: uniform semi-continuity implies uniform continuity. Thus the answer to my question is a resounding "no", since any function that is upper semi-continuous but not continuous cannot be uniformly upper semi-continuous.

Proof. Let $f$ be uniformly upper semi-continuous. Then for every $\epsilon>0$ there exists $\delta>0$ such that for every $x\in X$, we have $f(y) < f(x) + \epsilon$ whenever $y\in B(x,\delta)$. However, since this statement holds for every $x$, it also holds with $x$ and $y$ reversed; in the language of the original post, both $x$ and $y$ are contained in the set $X_\delta^\epsilon = X$. Since $y$ is in this set and $x\in B(y,\delta)$, we also have $f(x) < f(y) + \epsilon$, and thus $|f(x) - f(y)| < \epsilon$. But this is just the definition of uniform continuity.

$f(x)=0 \ (x\le0)$, $f(x)=-1/x \ (x\gt0)$ is upper semi-continuous on $[-1;1]$ — but not uniformly.