When a named Villager is converted to a Zombie Villager it will keep its name, but apparently not the persistant-tag, as it will despawn. This should be considered a bug as the player would expect named mobs to not be able to despawn. Also if a Zombie Villager is named after being turned, it will be persistant and not despawn, adding to the confusion, since both are identical to the naked eye.

To reproduce name a villager, let a zombie turn him and then fly 128 blocks away. The named Zombie Villager will now have despawned.