When is it okay to update other search committees? I'm on the academic job market this season for a position in a STEM field. I've had one on-campus interview and may receive an offer from this school within the next week or so. I've also had Skype interviews with a few other schools, but haven't received any on-campus invites from them. (The Skype interviews occurred about 3 weeks ago.) 

At this point, should I let these schools know that I've had an on-campus interview? Or is it safe to assume that I've been rejected by them if I haven't received a campus invite already?
Don't make assumptions. Nearly everywhere will let you know when you are no longer under consideration. You may or may not be high on their list. You could be high and they are just dithering. But they would rather make it definite, just to avoid fruitless communication. 

But it is probably a mistake to tell them you've had an interview, as it may confuse them about your interest. Wait until you have an acceptable offer before you notify them. 

Trying to "nudge" them in your favor could work either way, if it has any effect at all. It is a risky thing to do. Nudging might be worthwhile if you are juggling multiple offers.