I recall one position that was stalled until I followed up with the hiring manager, who was early into the organization and had long tenure and a lot of influence.
I reached them directly. They said they'd put the request in to HR. We hung up after the hiring manager said they would give HR a follow-up call. Within 24 hours, HR was calling me, asking how soon I could come in.
Most hiring managers don't have that much influence with HR (instant turnaround). But if they hare a good reputation internally, and HR is good at their job, then HR knows that such strong interest is reason to proceed expeditiously.
If HR is unresponsive to them, a hiring manager with budget -- an internal client of HR -- then maybe that is a place you don't want to work. Disfunctional HR tends to become a burden, over time. (Perhaps I should say, abnormally disfunctional HR; some would argue that HR is inherently at least somewhat disfunctional, from the perspective of those outside of HR and Legal.)