Version from Beth's Music Notes.
Off to Google Books, where I searched the "The Old Gray Cat Is Sleeping" and looked in the 19th Century. I had a hit from Google Books (
Michigan School Moderator, Volume 11) and found a poem called "The Little Mice" by Gertrude Walker.
I go and check out Songs and Games for Little Ones by Gertrude Walker, from 1887, and find the song titled "The Little Mice Are Creeping" and the composer/author listed as Margaret Bradford Morton. Archive Link. It seems like most of these melodies didn't make it to widespread use (see the adaptation of "Twinkle, Twinkle" below.
Margaret Bradford Morton appears to have been a composer of hymns and songs for children, as seen here, and here; and her name and Gertrude Walker's name can be found in this Women's Medical Journal, as it appears Margaret donated an ambulance in honor of her sister.
Amongs these resources, they mention a "Jenks Book, p. 87" as a source, but I could not find an online copy, until I searched a few more times, found a reference to "Walker and Jenks' Book" and then realized that that is the source I already found.
I attempted to try to find where/when the music switched to 6/8. Here is an alternate melody from 2005. I also found it in Alice Hammel's 2017 resource "Teaching Music to Students with Special Needs" with the same melody from Beth's Music Notes (above).
It's found in Lynn Kleiner's Book, "Kids Make Music, Babies Make Music, Too!: Teacher's Guide" with a slight variation in the last phrase.