The idea of a victim pressing or dropping charges is largely a myth. It is true that the law may take the victim's wishes into account for minor offenses but neither the police nor the prosecution service need their permission.
It'd be a terrible thing if this was the case- you'd give every criminal the incentive to intimidate or silence their victim.
In this case the police would continue to investigate and would charge Alex with some kind of assault (depending on the exact nature of the attack). Given the nature of the crime, the prosecution service could likely secure a conviction without Bob.