When you say that a link was approved, are you talking about the site you submitted it to, or approved by Google? I was also wondering how long it takes to get a backlink from sites like Digital Point or Flickr.
Usually this terminology refers to "Having your link approved by a Directory Editor" For example, when you submit your site to DMOZ (Open Directory Project) --- the most important web directory on this planet--- it usually takes about 3 months to 3 years to get your site "approved" by an editor. This variance depends on your category, the laziness of the DMOZ Editor, and the Backlog of Pending Sites in that category or niche. Typically, when you submit to Free Directories -- look at the "Pending Submissions" in the statistics box (lower left corner) as this will give you an indication of the Approval Time. For example --- XYZ Directory (I just made up the name) shows in their stats: 5,678 pending sites -- 67 sites approved today. By doing some simple math (5,678 divided by 67) -- you can see that it takes roughly 90 days for XYZ Directory to approve your link. Most directories try to TRICK the newbies by offering Instant Review for $$$. DO NOT waste your money! Eventually, you will get the approval e-mail as long as your site's not spam or porn, and you followed the instructions on the submit site form.
Once you have got approved your site will be listed in the directories or some other sites you have submitted.
lot of directories are moderated where admin/moderator review the submission before accepting the link tho the directory. once the directory admin approves the submission, you get an email saying your link has been approved. at that point, you will find your link on the directory under the category you submitted to. and if you can see the link, then there is a change that google will too