The sentence "Let me know when the task is completed." could have two meanings:
- Let me know the time that the task is supposed to be completed.
- Once the task is completed, let me know it.
Which of them makes more sense?
I don't agree that there is any ambiguity. The sentence
Let me know when the task is completed.
has your second meaning
Once the task is completed, let me know [it].
If you want to be told when the task is supposed to be finished, I suggest
Let me know when the task will be completed.