You rating lags a bit because Glicko2 uses your opponents’ games as part of your rating. The system looks back a bit at your opponents’ most recent rating trend (i think in the neighborhood of 20 games?) and uses that as part of the calculation.
Especially if your opponent has ongoing games, it can take a little time.
I think that’s about right, someone with more knowledge than I can probably give you a better explanation.
I find that somewhere between 2-5 minutes after completing a game if I do a forced refresh (hold down SHIFT while refreshing in most browsers) it will then update. I suspect that the score needs to be recalculated by the algorithm in its entirety.
After winning a game and returning to the home page(without refreshing) my rank has been updated on the left but not on the right.
I would agree with @lysnew that ‘This isn’t a big issue’ except that it may be responsible for confusing the issues around the ranking system by misleading some people as to the effect of any one game on their rank.