Peter Shor has put forward "A Wasp and a Bee" by Jane Taylor as a candidate for the verse about the bee and the wasp. For the other verse mentioned, about "the cat by the barndoor spinning" I propose the nursery rhyme Three Little Mice:
Three little mice sat down to spin
Pussy came by and she peeped in
What are you doing my little friends?
We're making coats for gentlemen
May I come in and bite off your threads?
Oh no gentle pussy, you'd
bite off our heads
Little mice you're very wise
I love your whiskers and big brown eyes
Your house is the prettiest house I've seen
I'm sure there is room for
both you and for me
The mice were so pleased, they opened the door
And pussy, she left them all dead on the floor.
The rhyme thus involves a cat and spinning (although it's the mice doing the spinning rather than the cat). Some
alternate versions have as the first line "Some little mice sat in a barn to spin", and so a barn is also mentioned. The rhyme dates from 1850, and so is roughly contemporary with "A Wasp and a Bee", making it plausible that the gardener could have encountered both verses.