I agree that a crying toddler is a pretty normal thing but they do have to consider the needs of other patients. Perhaps there were other patients there with a migraine or earache? I can't stand my own children crying when I have a migraine (and the one time I had earache as an adult, I pretty much hid in my bed for a few days!) so somebody else's child would be unbearable. If it was one adult with earache and one screeching child, maybe the adult could get up and wait outside but if there were a couple of adults who needed quiet, I don't see why a toddler's need to scream would take priority over everybody else's pain.
I would expect a receptionist to ask if the mother could control a boisterous child, but if they were screaming in pain they should suggest an alternative place to wait, perhaps an unused office or HV room. I wouldn't expect to be told to go outside, but going on what the PP said, it doesn't sound like the mother has been exactly truthful so that may well have happened!
The mother is quoted as saying, "'It's virtually impossible to tell a 16-month-old to sit down and be quiet". Hmm not sure I agree with that.