I thought that perhaps I could find out when the the plague nonsense entered children's books at the Library. I received a bit of a shock. Books by Greenaway and Opie, Lavender's Blue and other standards of my children's days and mine are no longer on the shelves. The oldest collection of rhymes, songs and play rounds was newly assembled and copyright in 1987: Christine Allison, "I'll Tell You a Story" (but she didn't). Only about half-a-dozen had rhymes of Rosie; there were two with Ashes, Ashes and the rest were a-tischoo or similar (no squats).
One definitely stated the plague source: June Yolen's Mother Goose Songbook, 1992, Carleton House. The others had no comment.
I couldn't check the books for the smallest children; paperback and plastic, they were binned in no order and packaged in transparent bags for easy pick-up by parents; it would have taken too much time and earned me too many disaproving looks to have tried to go through them.
I remember when classics like Winnie the Pooh and the Beatrix Potter and Burgess stories were reprinted over and over. Now, they have either been rewritten by Disney authors or dropped altogether.