What's possibly confusing about the 'Khartoum' thing is that from viewing the film alone, one might get the impression that the killing took place overnight upon Tom's departure...and that Woltz wakes up to find the head the very next morning!

In reality (fictional reality that is), a couple of days probably go by while Tom returns home, informs the Don of Woltz's refusal to cooperate, lets him know about the prized horse...and THEN Don Corleone sends somebody back to Hollywood with orders to make Mr. Woltz an offer he can't refuse.

Apple