It can help. As a hiring manager I've often been in the situation of having a candidate I like, but the machinery of approvals and some upper manager who demands to 'chat with all candidates' but of course he's never available is getting in my way.
In that case, if a candidate says they are considering another offer, it lets me:
- Set expectations with them about how much time is left on our side
- Go use that to try to kick others into quicker action (and that can work)
- At worst, tell you "hey you should probably take the other offer" and then at least you don't have ambiguity.
You don't have to be overly pushy, just email saying "I am really interested in the position there but I've received another offer and need to give them a reply soon. Is there any update you can share on an opportunity for us to work together?"
However, remember a bird in the hand is worth two in the bush - if A and B can’t give you an actual offer by the time you need it but are still tempting you with “maybe soon,” you may lose the offer with C and possibly end up with no offers. Don’t let yourself get out in a bad position.