You will no doubt have had the above conversation countless times. It’s become some kind of glib, I.T. in-joke. The response is generally accompanied by a raised eyebrow and knowing look; “It’s not my fault it’s been dumped on me from a great height”. And of course, they’re now dumping the problem on you. It’s a form of Tennis, but not a particularly great one.
So, in the absence of a time machine you can, by all means do the best you can to satisfy the request, but make it clear at the outset that you will not be held responsible for failure to deliver on time. Remember, you are helping them out. Someone else’s lack of forethought and planning does not become your problem unless you allow it to.
You can of course look to use the event to gain yourself some brownie points, perhaps even seeding the grapevine with stories of how you pulled them out of the fire.