Obviously, I agree with encouraging everybody to stick to a deadline. Given the volume of documents they might need to get translated, I don't know whether every department will be able to deliver on time.
I think writing a letter back that encourages everyone to meet the deadline is absolutely appropriate. It can say, “If you cannot, please produce everything you have on a rolling basis, as quickly as possible.” We can then assess whether anybody has not met their entire burden and come back a couple of weeks later and figure out what to do.
A polite letter saying, “We expect you to meet the deadline, and if you can't give us everything you have, keep producing them...rolling...as quickly as you can” is probably the fairest and most reasonable way.