The answer could be that Obi-Wan's exile, both physical and spiritual, on Tatooine prevented news from reaching him. On a practical level, Obi-Wan did not know that Anakin had survived simply because he was isolated on Tatooine.
What Obi Wan didn't know is that Anakin survived the aftermath of their vicious battle on Mustafar. Kenobi assumed Anakin perished as a result of that conflict. He wasn't on the planet to witness Palpatine show up to save his apprentice from succumbing to his battle wounds.
Obi-Wan's decision to spare Anakin was influenced by the dark feeling that accompanied Emperor Palpatine's arrival, preventing him from finishing him off. If Obi-Wan had killed Anakin, it would have drastically changed the course of the Empire and Luke Skywalker's journey as a Jedi.
Obi-Wan Kenobi was aware of the truth, of course. The Obi-Wan Kenobi Disney+ TV show revealed he didn't actually know Anakin survived their battle on Mustafar until a decade later, when he was drawn out of hiding by Reva and the Inquisitors.
There's no doubt that Anakin was among the most powerful Jedi in Star Wars, but Obi-Wan was wary of his padawan's obsession with victory. Obi-Wan didn't have the same control over Anakin as a Jedi knight is supposed to have on his padawan and was routinely caught in Anakin's daredevil antics.
What If Obi Wan and Ahsoka DIED in front of Anakin Skywalker
Why did Anakin stop trusting Obi-Wan?
Anakin Hated Obi-Wan Because of an Imagined Relationship With Padme. Although Padme and Obi-Wan's affair couldn't have been further from the truth, Anakin apparently built up a scenario in his mind, which was only heightened by the Jedi Council's obfuscation in other areas.
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.
Both Yoda and Obi Wan knew from the holo revelations in the Jedi Temple that Anakin was now Darth Vader. For some reason, Obi Wan did not realize Vader had survived Mustafar.
At first, Ahsoka has no idea that Anakin Skywalker and Darth Vader are one and the same. She discovers the truth in Star Wars Rebels when she encounters the Sith Lord in a space battle. What she senses is so shocking to her that she passes out.
During their confrontation on Mustafar, he was heartbroken seeing what happened to Anakin. Even when he had the chance to end him then and there, the Jedi Master couldn't bring himself to kill him. So instead, Obi-Wan left him limbless, not expecting Palpatine would come to rescue his fallen Padawan.
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.
Yes, when Darth Vader turned back to Anakin Skywalker and sacrifice himself to save his son, he dies and Obi-Wan uses the force to connect with Anakin's spirit. In that time Anakin apologizes to Obi-Wan, desperately asking him to forgive him for everything he has done to him and everybody over the years.
And all the way up until the third sister told him otherwise, Obi-Wan truly and completely believed that Anakin was dead. He believed that he died on the banks of Mustafar. And even if they knew for a fact the Anakin had survived, it comes back to the split personality.
When Qui-Gon Jinn, Count Dooku, or countless Jedi during Order 66 perished, their bodies simply fell because they didn't consciously become one with the Force as Obi-Wan, Yoda, or Palpatine did. As Anakin died, he was guided into the Force by Obi-Wan.
The sequence calls back to a similar scene in Revenge of the Sith, when Bail Organa orders Threepio's memory be erased to prevent him from sharing information about Anakin Skywalker's fall to the dark side.
C3PO was clearly created from some sort of kit or existing prototype since every protocol droid aside from coloring looks the same. 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.
Maybe R2-D2 didn't know Darth Vader was Luke's father because he didn't see Palpatine christening him Vader, or maybe he believed Anakin was dead because even Obi-Wan didn't know Anakin was alive.
Having been completely and utterly defeated during their first battle in Revenge of the Sith, Darth Vader harbored a grudge against his former master which fueled his hatred for many years. Obi-Wan Kenobi's third episode confirms Darth Vader's biggest mistake came when he had a chance to finish the job.
Darth Vader did not 'forget' he was Anakin. He believed he was not Anakin Skywalker; a completely different person whom he killed. You will read in EU materials, that he referred to his memories as Anakin, as 'The Jedi'. He suppressed most of Anakin Skywalker's memories - his memories while he was Anakin.
Padme Amidala's secret marriage with Anakin Skywalker was one of the biggest reasons he eventually turned to the dark side. His emotions and love for her often clouded his judgment, and sometimes led him to be violent.
