I know the phrase, of course, but I can't say that I've ever actually heard anyone say 'taken ill.' I'm not sure what you mean by a formal sertting. A dinner party, perhaps? Nope, I've never heard it said in any setting. In my opinion it's too haughty at worst and awkward at best. I've read it a few times. In the pharmacy you'd say simply that you're sick or your kid is sick. More precisely, we describe the symptoms. I hadn't thought about it but isn't that what people all over the world do? "Hi Doc, junior was up barfing all night after eating Aunt Erma's potato salad at our picnic yesterday. You got anything for him?"
Now that I think about it, I can think of times when it might be appropriate to say took ill. For example, at a funeral explaining to strangers or people you don't know well, what happened to the decedent. "He took ill on the flight and never recovered." It's not used a lot, in my opinion. Maybe others elsewhere in the world use it regularly. I really don't know.