Darth Vader's true identity as Anakin Skywalker was a highly guarded secret, known only to a select few individuals including Palpatine, Obi-Wan Kenobi, Luke, Han, Leia, R2-D2, C-3PO, Master Yoda, Bail Organa, Ahsoka Tano, Darth Maul, Grand Moff Tarkin, Grand Admiral Thrawn, Jocasta Nu, Ferren Barr, Verla, Reva, ...
R2-D2 was present when Padmé gave birth to Leia and Luke, so he knew Luke was Anakin's son – but what he didn't know was that Anakin became Darth Vader. In addition to that, Anakin's fall was surely a painful event in R2-D2's life, one that he definitely wouldn't like to remember or even talk about.
Very few people knew that Darth Vader and Anakin Skywalker were the same person up to his death in 5 ABY. Imperials who knew in the current canon: Sidious (emperor), Grand Moff Tarkin, all or most of the Inquisitors, Grand Admiral Thrawn, various clones in the early empire.
R2-D2 witnessed some of Anakin's best moments and knew him when he was an innocent boy on Tatooine; it might have been healing for Luke to hear these stories. Perhaps, like the stories about Padmé, they were simply too painful to share.
Darth Vader acts like he doesn't even know C-3PO or R2D2. They were with him since he was a child. He did everything he could to forget his past. Also, his plating had been replaced many times, and protocol droids all look very similar.
Throughout the three years of the Clone Wars, R2-D2 was Anakin's loyal helper and sidekick. In a daring rescue of Supreme Chancellor Palpatine, R2-D2 served as his master's assistant in his Jedi starfighter.
And while his warnings don't work, at the very least, it shows that Maul became aware of Anakin and what he would soon become. With the rise of the Empire, it's likely Maul caught on to Vader's existence pretty quickly.
Master Yoda knew that Darth Vader was once Anakin Skywalker - the boy he sensed too much fear in to train way back in Star Wars: Episode I - The Phantom Menace. For many years it's been assumed Master Yoda was aware from the day Vader gained his cybernetic enhancements and signature wheeze.
Serving Padmé Amidala. R2-D2 repairing the shield generator of Queen Amidala's Royal Starship. R2-D2 was built around prior or during 32 BBY. He was one of several astromech droids on Queen Padmé Amidala's Naboo Royal Starship during the invasion of Naboo by the Trade Federation that year.
Summary. Obi-Wan's seeming inability to remember R2-D2 in "A New Hope" was a deliberate choice to downplay the droid's arrival and avoid revealing too much too soon.
The Invasion of Naboo. R2-D2 repairs the shield generator of Queen Amidala's Royal Starship. In 32 BBY, R2-D2 was in the possession of the Royal Engineers of Naboo, and was assigned aboard Queen Amidala's Royal Starship as a maintenance droid.
Together, Palpatine and Vader commanded the Empire's army of stormtroopers. Vader lacked any kind of formal rank and was simply regarded as a "lord" by stormtroopers or any other Imperial. Imperial stormtroopers rallied behind Darth Vader, who they seemed to basically worship as the hero of the Empire.
Canon characters, such as Anakin Skywalker, whose relationship with R2-D2 is one of the best in the whole series, Rey, Poe Dameron, Hera Snydulla and Kazuda Xiono are fluent in the droid language. And after spending decades with R2 himself, Luke Skywalker is also able to understand him.
He didn't. Vader hit R2-D2, but that was not what he was aiming at. He was trying to shoot the X-Wing itself. In any case, he couldn't have possibly known that that particular X-Wing was carrying his old astromech droid.
He didn't recognize him because again, they all look the same & it was decades until he saw him again. There was no reason to believe he or R2 even existed anymore.
But Anakin remained staunchly against this idea for R2-D2, meaning the droid never had to worry about losing the memories of his past. As a result, R2-D2 could learn the value of intelligence, quick thinking and when and where not to speak up.
In Episode 3, he leaves Mustafar with an uninjured Obi-Wan, presuming Anakin dead. Prior to this, in the novelization, R2 mentions to C3PO that Anakin doesn't talk to him anymore. Also, it's possible this is the very last time R2-D2 ever sees Anakin himself.
