In the books, they were converted immediately only not active/moving. They had blue eyes, despite the fact that one ranger had brown eyes and the other had green eyes while they were alive. It is commented in the book that the wights remembered a bit of their former lives, since they moved straight towards Lord Commander's quarters after reanimating. I'll try to find and add quotes later.
From A Game of Thrones - Jon VII, using A Search Of Ice And Fire
-
"Othor," announced Ser Jaremy Rykker, "beyond a doubt. And this one was Jafer Flowers." He turned the corpse over with his foot, and the dead white face stared up at the overcast sky with blue, blue eyes. "They were Ben Stark's men, both of them."
-
Jon remembered Othor; he had been the one bellowing the bawdy song as the rangers rode out. His singing days were done. His flesh was blanched white as milk, everywhere but his hands. His hands were black like Jafer's. Blossoms of hard cracked blood decorated the mortal wounds that covered him like a rash, breast and groin and throat. Yet his eyes were still open. They stared up at the sky, blue as sapphires.
-
Dywen sucked at his wooden teeth. "Might be they didn't die here. Might be someone brought 'em and left 'em for us. A warning, as like." The old forester peered down suspiciously. "And might be I'm a fool, but I don't know that Othor never had no blue eyes afore."
Ser Jaremy looked startled. "Neither did Flowers," he blurted, turning to stare at the dead man.