When did Darth Vader learn that Luke took part in Leia's rescue from the Death Star?

While answering this, I ran into the following assertion by OP:

from what I can tell, Vader wasn't aware that Luke was in the Millennium Falcon when the Death Star captured it after blowing up Alderaan

I'm somewhat unclear on whether that's correct or not, so the question is:

1. 
   

   Was Vader aware that Luke was on the Millennium Falcon with Han and Obi-Wan during Leia's rescue?

   
   


2. 
   

   If he was, what was the point at which he became aware, and how? E.g., during the rescue/escape? At Yavin? At some later day?

   
   

Answer

There is no evidence that Vader knew Luke was aboard the Millennium Falcon when it was captured. To suggest that he should have been able to detect Luke's presence, since Luke was his son, makes no sense. If being near one of his children was all it took for Vader to notice them, he should have known Leia was his daughter as soon as he met her. As far as we know, this didn't happen.

The only thing Vader knew upon first seeing the Millennium Falcon was that it had a certain "guy who chopped off both my legs and one of my arms" vibe about it:

There were cameras and microphones in the detention center, and although the cameras were destroyed, the microphones were not:

Apparently there were audio... links between this detention station and elsewhere.
- Star Wars - Episode IV: A New Hope novelization, George Lucas and Alan Dean Foster

So there were audio recordings of everything that happened in the detention block, so - assuming that he was smart enough to check them after Leia escaped - Vader would have gotten all the information he needed from Luke's introduction:

[Luke, dressed as a storm trooper, opens the door to Leia's detention cell.]

Princess Leia Organa: Aren't you a little short for a storm trooper?

Luke Skywalker: Huh? Oh, the uniform. [removes his helmet] I'm Luke Skywalker. I'm here to rescue you!

Princess Leia Organa: You're who?

Luke Skywalker: I'm here to rescue you! Look, I've got your R2-unit and I'm here with Ben Kenobi!

Princess Leia Organa: Ben Kenobi? He's here!?

Note: Even if there hadn't been recordings from the detention center, there were presumably wireless communications scanners, which would have recorded Luke's chatter with C-3PO, as well as a recording from the hangar control room where Luke, Han, Obi Wan, Chewbacca, R2, and 3PO went after getting off the Falcon:

BEN I don’t think you boys can help. I must go alone.

HAN Whatever you say. I’ve done more than I bargained for on this trip already.

LUKE I want to go with you.

BEN Be patient, Luke. Stay and watch over the droids.

Obviously, Vader already knew that Obi Wan had been there (because he killed him), and he would have been able to piece together the rest of the story pretty quickly. A few days before the Millennium Falcon showed up, it had blasted its way through a blockade. Just before that happened, an old man in a robe had lopped off someone's arm with a lightsaber in the Cantina. The old man was with a blond kid. Days prior to this, the droids with the stolen Death Star plans had jettisoned from the consular ship and landed on Tatooine. All of this happened not far from the house where Vader's stepfather lived. The stepfather and his wife were dead, and Imperial stormtroopers had killed them. This was the Skywalker family, and obviously, Luke had been living with them.

Everything Vader needed to know, virtually presented on a silver platter.

As to when this happened, I would think that the first thing Vader did after the Falcon escaped (and after the people Vader chose as scapegoats had been Force strangled) was to find out how the Princess had gotten away. This probably happened within hours at the most. It probably would have taken another day or two for Vader to hear about what had happened on Tatooine.