The specific rule for two-handed weapons can be found on page 147 of the PHB:
Two-Handed. This weapon requires two hands to use.
Nowhere does it have any additional rules for creature size.
Does Enlarged make a difference?
There may be some guidance in the Enlarge/Reduce Spell (PHB, 237) with my emphasis:
This growth increases its size by one category -- from Medium to Large, for example. If there isn't enough room for the target to double its size, the creature or object attains the maximum possible size in the space available.
Until the spell ends, the target also has advantage on Strength checks and Strength saving throws. The target's weapons also grow to match its new size. While these weapons are enlarged, the target's attacks with them deal 1d4 extra damage.
While it doesn't state that the now larger creature can hold a two-handed weapon in one hand, it does give extra damage for the extra large weapon. The text suggests that it is the extra size of the weapon that deals the additional 1d4, and not that it is an extra large creature wielding it.
The extra damage delivered by the larger creature is due to the larger weapon wielded by the larger creature - not just the larger creature.
What this means for you: Size Doesn't Matter
As always, a DM can rule how they'd like, but it is fairly clear that the two-handed property limitation is not based on creature size, but on weapon property requirements. What matters is having both a larger creature AND a larger weapon for there to be an increase in damage or abilities.
Why just one-handed?
Andras' answer on is there a way for a medium creature to wield a two-handed weapon in one hand brings up a very valid point. If you're only doing this for fluff reasons and not as a means to dual-wield or have a shield, then there really is very little harm in doing so.
However, if he does not want Duelling or Dual Wielding benefits, and he is ok with keeping the other hand empty for "balancing", I see no problem allowing it.
At that point it has no influence on game mechanics, it is just aesthetics.
For NPCs, it's all about the stat-block
If it doesn't state that they can do something that isn't normally allowed, then they probably can't. Larger weapons with larger and stronger creatures generally deal more damage then the PC equivalent of Medium creatures with their associated sized weapons - but unless it says they can do something that would be an exception to the rules, then they can't (like use smaller weapons from other creatures differently than what the weapon properties state).
Weapons and their sizes are associated with the same creature's size.There isn't a mixing and matching of weapon size and creature size to increase capability.
Let's say you allowed the large creature to use smaller creature's two-handed weapons in one hand. Just as a heavy weapon wielded by a Small creature is done at disadvantage because it wasn't made for that size, a similar rule may be used in this case. The smaller weapon isn't made for the larger hand, and disadvantage may be used here as a balance. For every plus, there is often a minus to balance - this would be a possible resolution to the allowance.