This is my wife's great great grandmother. She has memories of being told about a "Lady Rhoda Coates". But so far my researches have turned up someone who was daughter of an agricultural laborer in Sussex, England who was married twice to agricultural laborers and twice widowed. And in 1911 she was living with a stepson who was a carter on a farm. So I haven't done a comprehensive study of the Coates line, but it doesn't look like there is a connection. One son, Amos was a farmer and another immigrated to Canada in 1913 where he worked as a laborer.
Wikipedia has this on the Coats company:https://en.wikipedia.org/wiki/Coats_Group
So the Coats thread people were Scottish, not English. And for a company that is the second-largest manufacturer of zips and fasteners in the world, pretty funny that their Chairman is named Mike Clasper :)