In short, it can vary depending on video length and how busy the YouTube servers are - this is taken directly from the YouTube support pages:
"An uploaded video will immediately be visible in an authenticated user's uploaded videos feed. However, the video will not be visible on YouTube until it has been processed and indexed. The length of time between when the video is uploaded and when the video is publicly visible on YouTube varies. However, videos are usually indexed within one day and may be indexed in several hours or even less than an hour."