I'm a first level Warlock, Hexblade archetype from Xanathar's Guide To Everything. My question is:
Would a weapon with the Versatile property work with the Hex Warrior ability without Pact of the Blade if I fight in two-handed mode?
The Hex Warrior feature says (XGtE 55-56) :
The influence of your patron also allows you to mystically channel
your will through a particular weapon. Whenever you finish a long
rest, you can touch one weapon that you are proficient with and that
lacks the two-handed property. When you attack with that weapon, you
can use your Charisma modifier, instead of Strength or Dexterity, for
the attack and damage rolls. This benefit lasts until you finish a
long rest. If you later gain the Pact of the Blade feature, this
benefit extends to every pact weapon you conjure with that feature, no
matter the weapon's type.
The Versatile weapon property says (PHB 147):
This weapon can be used with one or two hands. A damage
value in parentheses appears with the property—the damage when the
weapon is used with two hands to make a melee attack.
And the Two-Handed property says (PHB 146):
This weapon requires two hands when you attack with it.
Best Answer
Yes, the effect still applies
Using a weapon with two hands does not automatically give a weapon the "two-handed" property which is a specific property of certain weapons (SRD p.65):
Using a Versatile weapon with two hands simply changes the damage dealt by the weapon, it does not grant or remove properties from the weapon.
See also this tweet from Jeremy Crawford which supports this claim.