I set an enchant card that takes effect "when day changes to night", and my opponent set an enchant card that "sets the clock to midnight". At the start of the turn, the Chronos changed from night to day.
Does my enchant card's effect take effect?
If you're the night side player, it does.
The Chronos changed from night to day at the start of the turn, so effects are resolved starting with the day side player.
The card that "sets the clock to midnight" is resolved first, so the Chronos, which changed from night to day at the start of the turn, moves from day to night (midnight) by that card's effect.
Time has now gone night → day → night, so your card's effect, resolved after that, takes effect.
If you were on the day side, your card would be resolved first, but at that point day hasn't changed to night, so it finishes resolving without taking effect.