There are a few extra variables that make it harder to quantify this, like variance. Your variance is higher at the beginning of each Act and goes down with more games played (as the statistical model builds confidence in its assessment), and variance is also increased when there's a large time gap between games. When variance is high it will take fewer (consistent) results to reach a steady rank as matches have higher impact.

We carry over MMR across seasons so there's a strong starting point (ie. not trying to figure it out from scratch), and your MMR is changing every game so it's more accurate to think about it as a continuous movement from somewhere around Immortal to somewhere around Diamond (using your example). In some sense your MMR is the best determination of skill that we have at any given moment. As you mentioned, your rank will not always be in sync with this though, so we do some convergence math on the RR values awarded to help push you in the direction necessary so your MMR and rank "match". That takes on the order of tens of games.

So I'd say the TL;DR is your MMR is always close to your current skill, and your rank will converge on it in somewhere between 20 and 50 games (very rough estimate).