The Dangerous Dogs Act 1991 only applies if the dog is in a public place.This owner's home is not a 'public place'. That would be a criminal case.
The Dogs Act 1871 applies wherever the dog is.Procedure is by complaint in the Magistrates' Court.That's a civil, not a criminal, procedure against the owner of the dog.It's not the law, in either case, that a dog is allowed one free bite, but evidence that the dog is normally under control and has not hitherto indicated any dangerous tendencies should avail the owner.Magistrates can, in theory, order the destruction of a dog under the 1871 Act but that's highly unlikely.
The victim has, in any case, the ordinary civil remedy of suing for damages for the injury and any loss.
Calling the police in is a good idea because they will advise the owner about future conduct, even though their powers are very limited in this case. A few warning words, including reference to the noise nuisance, might bring results.
This dog was probably 'protecting its territory' in this instance (no consolation to the victim) and not 'dangerous' in general.It could do with proper training.In any case the owners ought to put a mesh guard around the letter box so the dog can't bite an 'intruding' hand as well as keep the dog under control.