When a druid shapechanges into an animal and the animal the druid changed into dies, the druid is dead. As far as I know, this is not how it should be according to 5th ed.
The druid should change back and only take the damage in his original form, which could not be absorbed by the HP of the beast. If this damage does not exceed the pre-transformation HP of his original form, then the druid should still be alive.
