Let's say that I move a villager from the overworld to the nether and place a bed, would he sleep on it?
Beds placed in the nether will explode when you try to use them. Fortunately for villagers, this rule doesn't apply to them and they can sleep in beds as usual when in the nether.
According to the Minecraft Wiki:
Villagers can sleep in the Nether or the End, if the Overworld's time is correct.
It can also be seen in this Reddit post: