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I am sorry to hear you are worried about your Westie's nose and breathing. My name is Rebecca. I am a veterinarian and will do my best to help.
How long has he been like this?
The bad smell from his breath could be loose or infected teeth and gums. The roots of the large back teeth go up into the sinuses in the nose, and can cause infection there. If he were mine, I would take a good look in his mouth at all his teeth, especially the back ones. Antibiotics may help, but if he is has loose or infected teeth, the problem will continue until the infected teeth fall out or are extracted.
This breed often has skin disease from allergies. It is possible that this problem is also related to allergies, but at his age I would suspect a tooth problem.
Does he have have bleeding from his nose? I hope not, as that can be a sign of a tumor in his nose, especially if the bleeding is from only one nostril.Let me know what else I can answer or help with.
I hope you were able to see my answer. Was this helpful? Do you have any other questions for me?