There is no hard and fast rule with any of this stuff, but you will probably never hear back at all. I think there may have been a time in which hiring managers confirmed that they received your application and told you when someone was hired as a courtesy. Perhaps this still exists in other industries, but for the vast majority of Hollywood assistant gigs, you will never hear anything unless you're brought in for an interview. Why? Because people get bombarded with dozens of resumes. Because people suck. I don't know. You may get a "thanks! Will pass along!" email if someone is collecting for a friend of a friend. But in many cases, you may not be submitting your stuff directly to the person who's hiring and won't be able to follow up with that person.
Sometimes job postings stay within a trusted group of 5 or 10 colleagues/friends, and in those cases, you may have more personal contact. But often, postings appear on tracking boards with hundreds of members, and all those members are forwarding to even more friends. What may seem very "insider" might end up as widely read as the UTA joblist.
If you DO hear back, it might be within hours or a day. I once interviewed for a job on a Thursday that would have started the following Monday. If the people hiring are a bit more flexible, it might be a week or two. After three weeks, I think it's safe to assume that you're not getting an interview.
Don't take this as a huge discouragement; just remember that there is a LOT of competition for jobs, and you may have to apply to a ton of them before you get any bites.
As someone who has sorted through emails and resumes, I recommend that you keep your cover email very short and be sure to follow the specific instructions in each posting.
More info on the topic:
Cover letter and email style for job and internship applications