How were dead men converted into wights weeks after encountering White Walkers? [duplicate]

Question

In the eighth episode of the first season, Jon Snow and Jeor Mormont come across two dead rangers. The rangers were dead for weeks and once accompanied Jon's uncle, Benjen Stark.

After Jon and Jeor bring the men's bodies back to Castle Black, one of the bodies reanimates into an ice zombie. Jon runs it through with a sword, but that eventually destroys it with fire.

Sam read that corpses touched by the White Walkers are reanimated and can only be killed by fire.

Why weren't the corpses converted to wights minutes after being killed by White Walkers? Why did it take them weeks to reanimate?

Edit to add: The other question linked to this one asks if dead bodies can become wights south of the wall, while this one asks if they become wights weeks after White Walkers touch them. That one is about the location of becoming a wight, and this one is about time since being killed and transformed by a White Walker. Close, but not a duplicate. :-)

Please give citations from the books if you can. I consider the books a higher canon than the TV series.

Answer 1

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.

Answer 2

This has not currently been explained definitively in the TV show and I cannot speak for the books, as i have not read them.

As seen in the TV show, S5E8 'Hardhome', the corpses of all of the dead wildlings and brothers of the watch do not reanimate until the Night King raises his hands to force them to. This suggests he can convert them at will.

It may be possible that they were intended not to reanimate until they were back at castle black in the warm, on purpose, as a form of 'sleeper' attack to cause chaos and kill some brothers of the watch, or perhaps until the Night King expressly commands them to rise, they do so on their own, albeit slower.

This is only a theory, however. It has not been explained by the showrunners, writers or actors.

In regards to your edit, maybe they can go past the wall before they are reanimated or if brought there intentionally by a human.