It will take anywhere from a few days to infinity to get listed. Unfortunately, there are no guarantees, no standard timeframe and not much you can do about it except submit and move on.
A link from DMOZ will help your rankings about as much as a link from any other large, reasonably well-respected directory will do, which is to say, not a whole lot.
More links from higher-quality sources are better than fewer, so getting a link from DMOZ is certainly not going to hurt your site. But if you're expecting some noticeable jump in rankings when/if a link to your site goes live, you're setting yourself up for disappointment.
Submit, following the guidelines as best you can. Then get on with your life and your other web marketing projects. Eventually your site will be listed (or maybe not). Either way, it's not worth spending any additional time over it.