How can the oath check upon resolution if it is no longer in play? Or perhaps because the enchantment is no longer in play, only the first trigger remains on the stack, and the second trigger never occurs to change the outcome of the trigger.
It can check on resolution because, as it was pointed out:
Once the ability is on the stack it exists independently of the source of the ability.
There is no "second trigger," unless I'm misunderstanding your question. There is only one triggered ability with Oath of Druids, it simply checks to see if the condition is true to trigger, and then the condition must still be true when it resolves or it is countered on resolution. This is very similar to the way the card Terror works. In order to place Terror on the stack, you have to have a legal non-black target. When Terror resolves, it checks to see if the creature is non-black, and if it is not Terror is countered.