This song was sung to me as a bedtime lullaby along with a couple of other old sad songs by my Grandmother who was born in 1900.
She sang it to her own children in the late 1920's and '30's. And continued singing to all her grandchildren born in the 1950's and 60's.   So I know it has to be the version from the 1800's.
Her father was born in Scotland - her mother in England - so I think it must have come from that area - as someone mentioned going to Ireland and people they talked to knew the song right away.
The tune she sang is different from all the recorded ones I have listened to so far and it wasn't quite as long as some of the lyrics listed. She called the song, "Two Little Children"
My Grandpa had her sing it for their Posterity when she was 80 years old - so I am lucky enough to have a recording of her singing it so I know the exact tune she sang the words to.
It is truly a blessing to have it. She lived to be 96 years old!!
The other 2 songs she always sang to us at bedtime were: "The Black Sheep" and "Hello Central, Give Me Heaven:. I loved when she sang them to me even though they were all quite sad. Hope that helps anyone looking for information on how old the song must be.