A possible solution to the question,
who wrote the little script that Andrews and Green speak between musical numbers,
which serves to link the many short compositions/rhymes together.

Have just been using Google searches to see what comes up if I search on this album and its performers.
And the name that comes up is a woman named Julie Laurence.
It seems that Julie Laurence was active in the New York theatre in the 1950's and 1960's, and her specialty was children, and children's literature. Not only did she
write the script,
she wrote the lyrics for the opening rhyme,
the closing rhyme ( a prayer, it seems)
and as for all those nursery rhymes:

Julie Laurence is the one who did the research, collected literally
dozens of children's texts and songs,
and chose which ones were to be
set to music by Louis Hardin a/k/a Moondog.