I'd never seen any part of it except the chorus before. So I never thought of it as a stammer, just alliteration.

Who sings it? - My Dad used to sing it to my Mum (Kathleen), and I that was almost certainly before 1959, but he would have been only a toddler in 1918, so maybe it was popular sometime in between.

It was one of the songs sung after Christmas dinner at Grandma's house, round the table, one for each person.