A bounded derivative is a sufficient condition for uniform continuity, but not necessary.
I know the counterexample $f(x) = \sqrt{x}$ on the interval $[0,\infty)$ where the derivative is unbounded at $0$, but the function is uniformly continuous.
Is there an example where $f$ is uniformly continuous and $f'(x)$ is unbounded as $x \to \infty$ but bounded on any compact interval?
Taking $$f(x)=\frac{\cos(x^3)}{x}$$ on $[1,\infty)$ suffices since it is uniformly continuous and has the following derivative:
$$f’(x)=-3x\sin(x^3)-\frac{\cos(x^3)}{x^2}.$$