Rulings

If an opponent has two or more cards in their hand as their upkeep begins, Davriel's first ability won't trigger. No player may take any actions during a turn before the turn's upkeep begins. Because the upkeep step is before the draw step, that player won't have drawn a card for the turn yet.
If an opponent has one or fewer cards in hand as Davriel's first ability triggers but has two or more cards in hand as that ability resolves, Davriel doesn't deal damage to them.
In a Two-Headed Giant game, Davriel's first ability triggers for each opponent separately as appropriate during that player's team's upkeep.