Did you try to exile the card in the upkeep or draw step? Or was it already the opponent's main phase? If it was the main phase, then as the active player they had priority to cast any kind of spell before you get a chance to do anything in that phase. With delve being part of the spell's cost, you simply had no priroity window to exile a card from the graveyard. You would have had priority earlier in the turn (upkeep step and draw step), and could have done your exile then. But that requires you to put a stop on your opponent's upkeep, otherwise Arena passes priority for you straight to the main phase as a shortcut. And you cannot interrupt the paying of a cost with the Cauldron.

Just because the opponent does not do something immediately, does not mean, that you can do your instant timing stuff at will. The game operates under the concept of priority, and even instant spells and activated abilities have to abide by that.