$\sqrt{x}$ in $[0, 1]$ is not Lipschitz continuous, yet is uniformly continuous.
$\sin(x^2)$ is neither Lipschitz nor uniformly continuous in $[1,\infty)$.
In first case, slope is horizontal, but the graph of the function is not exploding near $0$.
In 2nd case, value of sine is changing very drastically near infinity.
Can we make some generalizations here as to when non-Lipschitz continuous functions can still be uniformly continuous?
Any continuous function on a compact interval will be uniformly continuous as well; an example is $\sqrt{x}$ on $[0,1]$, as you mentioned. In general, if you can subdivide the domain into a finite number of pieces such that on each piece, either the function is lipschitz or the piece is compact, that would be enough to guarantee that the function is uniformly continuous. The example $\sin (x^2)$ you mentioned shows you can't do much bettern than that.