DM decision, but probably 2d8+1d4+STR damage at disadvantage
Lager weapon means more damage
The Ogre's Greatclub attack does 2d8 + 4 damage according to its statblock.
The rules in the DMG say that:
Big monsters typically wield oversized weapons that deal extra dice of
damage on a hit.
A creature has disadvantage on attack rolls with a weapon that is
sized for a larger attacker. You can rule that a weapon sized for an
attacker two or more sizes larger is too big for the creature to use
So the greatclub the orge is holding is extra big thus accounting for the damage.
A normal greatclub does 1d8 damage, but this oversized greatclub does 2d8 damage.
Enlarging the PC and weapon
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.
When a medium-sized PC carrying a large-sized weapon is enlarged one of two things will happen. The DM will have to decide which one is the interpretation that they prefer since there is some ambiguity here.
Large PC + Large greatclub
Since enlarge says the "weapons also grow to match its new size" one could interpret this as meaning that since the greatclub was already large that it does not grow at all (since large "matches" the PC's new size). Since the greatclub did not technically grow in size, it does not get the bonus damage from enlarge either. On the other hand, since the PC and weapon size now match, there is no disadvantage.
An enlarged PC using a large greatclub would do 2d8+STR damage (no disadvantage)
Large PC + Huge greatclub
One could also argue that this spell was not written with mismatched character/weapon sizes in mind and that the intent of the spell is probably that everything just increases in size proportionately. In this case the character and weapon both increase one size category. Since their size stays mismatched the character will still have disadvantage wielding the huge greatclub.
An enlarged PC using a huge greatclub might do 2d8+1d4+STR damage (with disadvantage)
Option #1 vs #2: bigger isn't always better
Given that the difference between the two options is a relatively small 1d4 damage, I personally would rule for whichever the players found more fun and apply that consistently.
Both options are pretty close in terms of damage output: option #2 would average to 13+STR (5+5+3) but will have disadvantage and option #1 would average out to 10+STR (5+5) and will not have disadvantage. #1 has the benefits of being more in line with the literal wording of the spell effect as well as not having disadvantage which can reduce damage output. Either way you choose, it shouldn't make a huge difference in terms of power.
Warning for DM
It is worth noting that monster-creation math and PC-creation math is intentionally different/incompatible with each other. As a DM be careful when you give a PC a monster weapon that you are giving them only the weapon part of the damage without any of the bonuses that comes from the monster itself. It is very clear in the ogre example, but may not be so clear with other examples.