After hearing that Hammond has called out sick, Dr. Armbruster comments that he'll check on Hammond during lunch. During the visit, he suggests Hammond be to work in about an hour. Lunch generally starts at 11AM and assuming even only a 10 minute commute, that would have Hammond arriving at the hospital at 12:20PM at the earliest. Yet after arriving, everyone was telling him "Good morning."
(Possibly deliberate.) The Hammonds' cat has a male name, "Seymour", but its patches of orange, black and white fur are hallmarks of a calico cat. Virtually all calico cats are female.
When Chris (Dr. Hammond) is delivering the baby, he picks it up immediately after the birth to reveal that the umbilical cord has already healed, and the baby is perfectly clean and dry.
Multiple members of the crew are reflected off the glazed pictures on the wall when the cleaning lady comes to tell Dudley Moore he is going out for groceries.
When the cat and dog were swapped, the dog was barking despite being in the cat's body and the cat was meowing despite being in the dog's body. The same should have happened with the voices of father and son when they swapped places.