The song is - as mentioned by IanC - based on an old English proverb/poem/rhyme (whatever you might call it) and I also think that S&G made a beautiful song of it

APRIL COME SHE WILL

April,
come she will
When streams are ripe and swelled with rain;
May,
She will stay,
Resting in my arms again.

June,
she'll change her tune,
In restless walks she'll prowl the night;
July,
She will fly
And give no warning of her flight.

August,
die she must,
The autumn winds blow chilly and cold;
September
I'll remember,
A love once new has now grown old.

THE CUCKOO
In April come he will
In May he sings all day
In June he'll change his tune
In July he'll fly
In August go he must