Depends on the directory (of course). My ezine articles are typically indexed in 24 hours, Go articles in 72 hours and other directories take more or less time based on how high their home page PR is (I know this isn't the real reason, just a good approximation) and if they are human edited or not. I will typically publish my articles from each site on a dummy blog to help indexing if I'm having any concerns.
It depends. Sometimes it can happen pretty quickly as others have pointed out. The more popular and higher PR the site, chances are that it will be indexed sooner, since Google crawls those sites more often than others.
Hey man it all depends when a Google bot visits and spiders the page. You can of course just send a ping to Google and blam! It's done
So I agree with the rest in that it depends on the individual directory's page rank and whether or not you are an expert author (ezine articles approve articles fasters from authors who write frequently). When your article is approved on ezine or anywere else and you have the url, tweet it up! Hopefully it gets retweeted and spreads!
You can ping your article to try to get it indexed faster, works fine for me. But if it's on a major directory you will not need to worry too much about it.