According to the Legends novel Dark Lord: The Rise of Darth Vader, Obi-Wan learned from the HoloNet that Vader was still very much alive, a few months after settling on Tatooine.
When Obi-Wan lifted his gaze, the intermittently garbled HoloNet was
displaying an image of someone outfitted in what almost seemed a
costume of head-to-toe black. Human or humanoid-the being’s species
wasn’t mentioned-the masked Imperial had apparently played a role in
tracking down and executing the “insurrectionist” Jedi, and enslaving
their Wookiee confederates.
He gets additional confirmation from Wuher, the Cantina Owner
The man adopted a conspiratorial voice. “You want to remain all right,
you’ll keep your voice down about Vader, understand? You’ll keep from
asking questions about him, too. Even in this Force-forsaken place.”
Obi-Wan studied him. “What do you know about him?”
“Just this: I have a friend, a trader in hardwoods, who was on
Kashyyyk when the Imperials launched their attack on a place called
Kachirho. I guess he was lucky to get his ship raised and jumped. But
he claims he got a glimpse of this guy Vader, ripping into Wookiees
like they were stuffed toys, and going to lightsabers with the Jedi
who were onworld. ” The spaceport worker glanced furtively around the
cantina. “This Vader, he toasted Kashyyyk, friend. From what my friend
says, it’ll be years before a piece of wroshyr goes up the well.”
“And the Wookiees?” Obi-Wan said.
The stranger shrugged forlornly. “Anyone’s guess.” Placing a few
credits on the table, he stood up. “Take care of yourself. These
desert wastes aren’t as remote as you may think they are.”
When the water arrived, Obi-Wan downed it in a gulp, shouldered his
rucksack, and left the cool shade of the veranda for the harsh light
of Anchorhead’s principal street. He moved in a daze that had little
to do with the glare or the heat.
As impossible as it seemed, Anakin had survived Mustafar and had
resumed the Sith title of Darth Vader. How could Obi-Wan have been so
foolish as to bring Luke here, of all worlds? Anakin’s homeworld, the
grave of his mother, the home of his only family members…
Shortly afterwards he gets confirmation from Qui-Gon's ghost that Anakin = Darth Vader (noting that this somewhat conflicts with the canon where he learns Anakin's new name before confronting him on Mustafar. Presumably he means this Darth Vader)
“Master, is Darth Vader Anakin?” he asked after a moment.
Yes. Although the Anakin you and I knew is imprisoned by the dark
side.
