Let's say the caster uses Enhance Ability and chooses Cat's Grace on themselves and rolls a 13 and a 16 with advantage on initiative, so they take 16. If the caster then loses Enhance Ability in-combat, or ends concentration on that spell, does their initiative roll revert to 13?
You already made the Dexterity check; its result does not change
The enhance ability spell states:
[...] The target has advantage on Dexterity checks. [...]
And the section on Initiative states (emphasis mine):
[...] When combat starts, every participant makes a Dexterity check to determine their place in the initiative order. [...]
[...] The DM ranks the combatants in order from the one with the highest Dexterity check total to the one with the lowest. This is the order (called the initiative order) in which they act during each round. The initiative order remains the same from round to round. [...]
Outside of initiative explicitly remaining the same, this also just makes sense. The game does not tell you to record the rolls you make when rolling with (dis)advantage so it should not expect you to remember a result from quite possibly hours ago (real-time). The result of a roll you already made does not change by new modifiers being added. Similarly, casting enhance ability on somebody during combat does not suddenly allow them to reroll their initiative with advantage.