A week wait is rather unrealistic. An animals health can worsen quite quickly. If a urinary tract infection is left alone for too long, it can cause a more severe infection in the dog, often moving to other organs and causing the dog to become very ill.

Even if he is peeing a lot, the animal still requires water so he doesn't get dehydrated.

If your vet won't move up your appointment, you should seek out another vet who can see your puppy right away. Treatment should not be delayed.