From a certain point of view Vader is correct. Rebel transmissions were beamed to the ship Leia's ship was inside, making it but an escape ship as far as Vader is concerned. My guess if Leia hoped they could slip away into hyperspace and use not only her diplomatic immunity but also that she has a common ship type to blend in when reaching Tattooine. No one could guess that Lord Vader could track her in hyperspace and come out of lightspeed almost right on top of her ship.
In the Radio Drama from a long time ago, Leia's ship jumps into the system to get the plans from a blockaded planet, using the excuse that they need to repair their navigation system of something like that. With R2-D2 on the outside to make it look good, and C-3PO just inside the airlock to keep R2 informed. Both droid are given specific orders by Captain Antilles and Princess Leia. They are to deny knowledge of her to anyone. They are on a classified mission, and Leia extends her order to R2 alone when she gives him the plans and goal of finding Obi-wan. Recall that on of the first things that 3PO says while still in the corvette is "There will be no escape for the Princess this time" meaning they've had some adventures with her before. When Luke asks if they know of the Rebellion against the Empire, 3PO simply responds, with more or less 'how else do you think we came into your service'. When asked if they have been in many battles, 3PO says 'several I think.' 3PO is being deceptive in the film, and lies quite a bit really, or at the least doesn't say everything. It doesn't seem like it because of the butler voice, but probably would have been more obvious if he's had the "slick used car salesman" voice Lucas had wanted at first. But Anthony Daniels made the role his, and it stuck. I suspect that R2 had the hyperdrive coordinates for Yavin IV for Han Solo once Leia was back.
Now the two droids are still in every single Star Wars film (aside from the two Made for TV Ewok movies).
For a locked down world, Jedha seems pretty easy to get in and out of. I image Dr. Evazan left not long after we saw them, as it would have taken time to walk all the way to Saw's place from the city, and it was some time before the Star Destroyer left. It was also still a little bit before the Death Star fired, so a ship or two could have escaped off camera easily. It isn't there story anyway, so they don't matter too much. I did notice that Saw had some X-wings parked outside