I meant to add that I'm sorry, I don't know any May Day songs or rhymes for children. I suppose "Here We Go 'Round The Mulberry Bush" wouldn't count. I'm not being facetious about that-i guess it depends on the age group of children you're talking about. "The Mulberry Bush" would be for younger children.

I'm also not being facetious about this suggestion:

I'd Like To Teach The World To Sing
[The New Seekers]

I'd like to build a world a home
and furnish it with love.
Grow apple trees and honey bees
and snow white turtle doves.

I'd like to teach the world to sing
in perfect harmony.
I'd like to hold it in my arms,
and keep it company

I'd like to see the world for once
all standing hand in hand.
And hear them echo through the hills
for peace throughout the land.

It's the real thing
what the world wants today,
That's the way it'll stay
with the real thing.

I'd like to teach the world to sing
in perfect harmony.
A song of peace that echoes on
and never goes away.

Put your hand in my hand
let's begin today,
With your hand in my hand
help me find a way.

I'd like to see the world for once
all standing hand in hand.
And hear them echo through the hills
for peace throughout the land.

I'd like to teach the world to sing,
in perfect harmony.
A song of peace that echos on,
and never goes away.

http://www.metrolyrics.com/id-like-to-teach-the-world-to-sing-lyrics-new-seekers.html

Here's a link to the YouTube video of this song:

http://www.youtube.com/watch?v=7h5ZmyzRRuU

"I'd Like to Teach the World to Sing (In Perfect Harmony)" is a pop song which originated as an advertising jingle, produced by Billy Davis and sung by the Hillside Singers, for Coca-Cola, and was featured in 1971 as a TV commercial.

The New Seekers also had a hit with the song around the same time"...
http://en.wikipedia.org/wiki/I'd_Like_to_Teach_the_World_to_Sing

Even so, I think it would be a great song to teach children 5 years of age and up, for May Day or any day.