Yes, this is well-known in the UK too. Just a couple of words difference:
Pat-a-cake, pat-a-cake, baker's man,
Bake me a cake as fast as you can,
Pat it and prick it and mark it with a B,
And put it in the oven for baby and me.
I liked the layout of this one Cathy. Peegue had to take the long way home so we got the rhyme again, which kids love.
