When does the time window start for the term "due"?
Solution 1:
Based on your revised question...
Generally, one doesn't refer to a task as being "due" without including a date. There are several statuses one might use for a task ("planned","in progress","complete","overdue"), but "due" isn't generally one of them. Instead, one refers to a task as being "due on March 1" meaning it should be completed by March 1.
For the "this task is due two weeks out" status, I would use "due soon" rather than "due."
Solution 2:
The multiple definitions of "due" make several of you right.
Merriam-Webster Online has as its first definition "owed or owing." So as soon as the service is rendered, payment is "due."
The fifth definition is "having reached the date at which payment is required." Under that definition, the bill becomes "due" on May 1. (I believe that general practice is to consider the payment "overdue" if it has not been received by close of business on the date shown as the "due date".)