Once a site is submitted to a directory, it has to undergo a review process before being approved. The process of review and approval of a site could take anywhere between 3-90 days after submission. If your site has been approved, it could take a few weeks to a few months before the search engines detect your links, since directories are very vast and have a number of categories and pages under them.
If you would like a rough estimation of the amount of incoming links from the directories to your site, you could type the following command into Google's search box:
+"abc.com"<space>+directory -site:www.abc.com (where www.abc.com is your URL).
The results that appear are those pages besides your own, that have the word 'directory' and your URL mentioned in them.