While humanity suffers and dies, Burton feasts on the corpses and easily evades efforts by humans to destroy him. Having eaten to his full, he decides to sleep in a vacant crypt. His sleep is so sound he winds up napping for an untold number of years. When Burton awakens, the world looks quite different to him.
Burton soon finds a suitable victim - a man who appears quite corpulent. Burton stalks him until he can isolate him on his own. Burton attacks the large man, but the intended victim shows no fear in the face of the vampire. Burton sinks his fangs into the man's neck, to no effect. It seems Burton slept far, far too long. The atomic war wound up eliminating all human life on the planet:
Robots:1 Vampires: 0