If someone told me on April 30th that she'd complete the project "later this month," I would presume she'd have it done by the end of May, not by the end of the day.
Sure, "within a month" would be a much more accurate way to phrase it, but I doubt I'd be confused over her meaning, unless this coworker was already known for such dry humor in her everyday speech, and I suspected she was hinting that the work was already just about complete.
The primary meaning of month regards a calendar month, but a secondary meaning is "a period of about four weeks." Many monthly cycles don't start on the first of the month, such as billing cycles and lunar cycles.
In writing, I'd fix it; that is, I'd change "later this month" to "within a month." Conversationally, though, I don't think I'd be thrown.
That said, in statement #2 ("later this month," spoken on the 15th), I'd probably assume that the word this signified we are talking about a period of about two weeks, not four – that is, until the end of the calendar month – but there could be some exceptions (such as if I was talking to my credit card company, whose months always seem to start and end on the 10th of every calendar month).
Most of the potential for ambiguity in these examples is found in the word this, as James pointed out above. We encounter the same potential for ambiguity when we say "later this week," or "later this year."