Thanks, Scribble. That came out as “Oh Playmate.”
There’s a parallel here, with the old Baptist hymn “In the Garden.” My Dad used to sing that, accompanying himself on the guitar. When I found the title and got sheet music for it I found I could not play it because I always broke down and cried. (He died 20 years ago.) 
We always sang “Oh little playmate” etc. (NY metro suburbs, mid to late 70s)
If I’m not mistaken Diane Keaton sings this in Reds. My son’s great-grandmother (b. 1915) delighted in teaching it to little ones and said she sang it with her school friends.
We sang “oh little playmate” etc. who “slid down my rain barrel into my cellar door”.
We also sang–
Oh little enemy
Don’t play with me no more
Don’t bring your dollies three
Stay off my apple tree
Don’t slide my rain barrel
into my cellar door
And we’ll be en-e-mies
forever more-more-mo-m–m’more.
Both versions were hand clapping games.
Western NY, mid-70s
Another vote for, “Say, say, oh playmate…”
Who is CeCe?
I also heard it as “Slide down my rainspout…”
That makes more sense than trying to slide down a rain barrel.
I grew up in New England and we sang “Oh Jolly Playmate” vs. Cee Cee, et al.
As for the second verse, I learned it as:
“My dolly has the flu
She threw up in my shoe . . .”
We were a classy neighborhood.
And the purpose of it was either
• to jump rope to; or, more often,
• to do that thing with the hand-clapping, where you alternate clapping your own hands, clapping the other person’s hands, and clapping your knees.
Yup - it’s true. We used to do that one, and:
Miss Mary Mack (Mack, Mack)
All dressed in black (black, black)
With silver buttons (buttons, buttons)
All down her back (back, back)
She asked her mother (. . . etc)
For fifty cents
To see the elephants
Jump over the fence
They jumped so high
They reached the sky
And never came down
Till the 4th of July
We did that one, as well as this for jump-rope:
Not last night
But the night before
24 robbers
Came knocking at my door
As I went out (jump out of the jump-rope)
To let them in (jump back in)
They hit me over the head with a bottle of gin
I asked them what they wanted
And this is what they said
Spanish dancers, do the splits (do the splits)
Spanish dancers, do the kicks (kick)
Spanish dancers, turn around (turn around)
Spanish dancers, touch the ground (touch)
I think there’s more but that’s all I remember, it’s been about 40 years. 