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
until ios5 you could not scroll internal divs - so you probably are not seeing a scroll bar when you try to scroll because there isn't one.
I haven't tested on ios5 but supposedly scrolling internal divs now works.
If it isn't an internal div then you should be able to see the scroll bar when it is scrolling only - this isn't just on ios anymore - lion has gotten rid of all native scroll bars too. You can only see them when a window is scrolling or when the window is first loaded.