In short: two weeks' overdue is uncommon at best, and a mistake at worst. Most people would consider it a mistake.
A longer analysis follows.
This is not necessarily a mistake. Arguably, it's a matter of style—and different style guides may give different guidance. (Although I've yet to see one that gives guidance in favour of this specific construction.)
Time periods are sometimes put into possessive form, to express the duration of or time associated with the modified noun:
the Hundred Years' War
a day's pay
two weeks' notice
The University of Calgary:
Use the apostrophe in phrases of time or measurement.
Examples: seven o'clock a month's leave my money's worth
Grammar and Style in British English: A Comprehensive Guide for Students, Writers and Academics:
The possessive apostrophe is also used for measurements of time –
a week’s time
two weeks’ time
six months’ jail
twenty years’ service
Apostrophes are used in time expressions (e.g., three years' experience, two days' pay, one day's time). These are also known as temporal expressions.
Having said that, note that there is a syntactic difference between all of the style guide examples above and the actual phrase in the question itself.
In the style guide examples, the possessives are used with nouns. However, the question's phrase is adjectival. So, can we use a possessive form with adjectival time-based phrases specifically?
The Grammar Monster reference above says that apostrophes are used in temporal expressions. According to Temporal Expressions (PDF), from the MIT Computer Science and Artificial Intelligence Lab:
When marked, the full extent of a temporal expression must be one of noun (4), noun phrase (5), adjective (6), adjectival phrase (7), adverb (8), or adverbial phrase (9).
Unfortunately, while that paper shows that a time expression can be adjectival, it doesn't give an example of the possessive being used.
Turning to Google Books Ngram Viewer, I ran a query with the following search term: weeks' time,weeks time,weeks overdue,weeks' overdue,weeks' late,weeks late.
- The most common time expression (noun) is weeks' time, with an apostrophe.
- The expression weeks' overdue (adjective), doesn't appear at all.
- The expression weeks' late (adjective) does appear, albeit with far less frequency than weeks late.)
What I would take away from this is that, barring a style guide that says something specifically different about the adjectival use of time expressions, the predominantly common styling is to omit the apostrophe from the phrase in question.
As such, two weeks' overdue may, arguably, not be a theoretical mistake—but it certainly looks like a practical mistake.
There is also a grammatical analysis that indicates the possessive with an adjectival phrase (or at least the one in the question) is actually wrong.
The possessive is a shortened version of the preposition of:
a day's pay
→ ✔ a pay of a day.
a week's time
→ ✔ a time of a week
But we can't use a similar substitution with the following:
two week's overdue
→ ✘ overdue of a week
If we were to use a preposition, it would be by—but the apostrophe doesn't represent that.