I've always known it as

Ladybird, Ladybird fly away home,
Your house is on fire and your children are gone...

Which is if anything, rather worse than just being in danger of burning. And that's how the Opies have it the Oxford Dictionary of Nursery Rhymes - with a note saying (among other things) "The rhyme is undoubtedly a relic of something once possessed of an awful significance" - but that might be tongue in cheek.

But I've never hear of anyone using this as a lullaby. It's what you say to a ladybird to encourage it to fly off to a safe place when its somewhere where it might get hurt.

Incidentally, why do Americans call them "ladybugs"? After all, they aren't bugs any more than they are birds, they are beetles.