Aiko

A few sneezes straight after a bath is usually water or shampoo up the nose. If he develops coughing, persistent sneezing, runny eyes or nose or goes lethargic over the next few days, that is more likely something he caught at the park, and either way it is a vet visit rather than a bath question.

Your mother is wrong on the mechanism and half right on the worry. A bath does not give a dog a cold: respiratory infections come from viruses and bacteria, usually picked up off other dogs, and being wet does not create them.

But a dog that gets genuinely chilled is a separate problem, and a whippet is near the top of the list for it. Thin coat, low body fat, loses heat fast. Shivering for an hour indoors is longer than I would want.