We'll go gathering nuts in May

[Hellyon]

Surely this doesn't make sense - nuts are gathered in autumn - so where does the saying come from?

[heathfield]

A popular explanation is that 'nuts' is a corruption of 'knots', and the song refers to gathering bunches of flowers from the hawthorn tree.
Personally, I like to think of it as just a nonsense rhyme, the sort of childrens' song that's always been around.