Why does the apostrophe go on the end of the following: Mothers' Union? Personally I would have placed it before the "s." For example: Mother's Union.
Mother's indicates possession by one mother. Mothers' indicates possession by more than one mother.
If the name were a shortening of something like "The Union for Mothers", (or "of Mothers" it might be known as the Mothers Union.
The Union of Plumbers might be called the "Plumbers' Union" or the "Plumbers Union", but would probably not be called the "Plumber's Union". (Because a union is a union of a plurality.)