I wonder how long that dragon cub is already stuck on that cliff.
lets assume not everyone who wanted to make contact with the MC immediately ran after they heard about the countess, in addition the info likely did not reach everyone at the same time. so that big stack of papers likely got collected over several days. worst case is probably a week or so.
also: who makes an emergency request targeting only a certain single helper. if they were truly interested in saving the dragon cub they would likely asked several groups / some emergency service.