None of the Imperial Senators in the Rebel Alliance council mentioned being freshly out of a job in Rogue One, so presumably sometime between Rogue One and A New Hope is when the Emperor dissolves the Senate. Thus, when she leaves Scarif, Leia has certain powers and privileges; but by the time Vader catches her, they are gone.
I can almost see them disbanding the entire Senate just to strip Leia of her diplomatic status once they realize it was her ship that took off with the data.
The Senate is dissolved during ANH.
Or the characters we see on screen are just learning about it, the timing could go either way.
Edited by MaxKilljoy