>I hope the submissions are reviewed in the order they were sent.
They usually aren't.
>How long does it take for a site that is reviewed and approved to show up in the directory?
"NORMALLY" (all daemons behaving, all servers up, no software upgrades with glitches -- in other words, do not bet money you need on it)
1) dmoz.org public pages -- 1 minute to 7 days;
2) RDF (published Open Directory Database) -- 4 to 10 days
3) google.com and AOL.com DIRECTORIES: 2 months "or so"
4) google.com SEARCH: several weeks at most (after step 2)
Note that all of these are automatic processes, programmed fallibly and run intermittantly by other people. This is all based on our prior experience or rumors of other people's activity. So if it doesn't happen this way, don't ask us why, and don't blame us!