Is there a general answer to give? How many days or weeks does it approximately take to get a proposal with 100% commitment into beat? What work has to be done by SE before the private beta can be launched?
When a proposal reaches 100%, the site typically enters private beta the following week on a Tuesday or Wednesday evening (Eastern Time). Technical delays, scheduling conflicts, or larger internal discussions can push out a launch date, but the vast majority of sites will typically launch in that first week or two.
The general answer would be, that in case of proposal in English language (this includes Language X & Usage) they are started as Robert Cartaino have stated, within one week.
But no proposal (AFAIK) in language other than English was started yet, and unfortunately, no one from the SE team cared to answer my question about the status of that proposals: https://meta.stackoverflow.com/questions/169728/what-is-the-status-of-support-for-foreign-language-ses
So the answer could be: from 1 day to 10 years ;)