No, it doesn't apply special ammunition effects to the area damage.
Hail of Thorns' description states that in addition to the normal effect of the attack the creature has to make a Dex save, taking 1d10 damage on a failure.
The next time you hit a creature with a ranged weapon attack before the spell ends, this spell creates a rain of thorns that sprouts from your ranged weapon or ammunition. In addition to the normal effect of the attack, the target of the attack and each creature within 5 feet of it must make a Dexterity saving throw. A creature takes 1d10 piercing damage on a failed save, or half as much damage on a successful one.
Arrow of Slaying says that a creature only takes the extra damage if it takes damage from the arrow.
If a creature belonging to the type, race, or group associated with an arrow of slaying takes damage from the arrow, the creature must make a DC 17 Constitution saving throw, taking an extra 6d10 piercing damage on a failed save, or half as much extra damage on a successful one.
The same applies to the others types of ammo. The extra damage from the Hail of Thorns spell doesn't count as damage from the arrow, but from a separate spell, that activates when the arrow hits; so the arrow's special effects don't apply to that damage.