There is no required waiting period between a dog biting someone and it being destroyed.
The sad fact is that this kind of situation is depressing common. People bring their pets in to get put down, quite frequently, for reasons like 'it's peeing on the carpets', 'we're going on holiday and it's too expensive to board it' or 'we don't want it'. Of course vets can refuse, and many do, but the animal is going to be destroyed no matter whether a vet does it or they do it themselves.
Someone arriving with a dog that has apparently bitten someone is usually not questioned as this is a 'valid' reason for euthanasia - but people can and do abuse this.
Julie, have you found out what happened to your dog yet?