This is not about what my mother sang to me, because I don't remember. But when my oldest son was a baby, I'd sing to him:

Rock-a-bye Teddy, you are a beaut!
You'll fill your pants, a rooty-toot-toot!
You'll fill your pants, and wet them also

And into the pail the diaper will go!

Rock-a-bye Teddy, in the white chair
When you are hungry, mommy is there.
When you've got pants-full, she'll be there too!
When there's diaper rash, she'll Vaseline you!

Both Theodore his younger sister Monika are adopted, and in equity I had to do something just for her, as follows:

Rock-a-bye Monika, you are a queen!
When you joined our family, we thought it was keen!
When we picked you up, little Teddy went too,
Now there's daddy and mommy, Teddy and you!

When, late and unlooked for, our son Hans was born to us, I had run out of rock-a-bye variations, but surely there had to be a song for him. He was breast-fed, and I sang him this. I don't have a way to pass the tune I wrote to you, but you have my (im)modest assurance that it's a lovely, haunting melody.

Go to sleep, my little boy.
Dream about a milk-filled booby!
Mama nurse you in a while,
When your time has come.

When your time has come, my boy,
Mama-milk there'll be in plenty
When your mama has let down,
And nursin' time has come.

When nursin' time has come, my boy,
Mama-milk will fill your tummy.
With the nipple in your mouth
You'll never need your thumb!

Dave Oesterreich