The jQuery time-picker plugin that I wrote uses a div as the containing block for the list of times, and on Mobile Safari there are no scrollbars to indicate that there are
By convention, scrollbars are not used on iOS.
For a div with overflow: scroll, the only native way to scroll is with two fingers.
You might take a look at iScroll, a JavaScript library which handles touch events and implements single-finger momentum scrolling (what users generally expect in native apps) for divs.