Here's the text copied from the Horatio Alger book at Project Gutenberg. I wouldn't call it "more or less identical" mainly because, while searching for it, I tried "rum see ho", "Tyburn Hill", "Newgate Stairs", "as I was going" and "beggarman"—all to no avail. I finally found it when I searched for "beggar man"!

"A beggar man laid himself down to sleep,
      Rumsty Ho! rumsty Ho!
A beggar man laid himself down to sleep
By the banks of the Mersey, so high and steep,
      Rumsty Ho! rumsty Ho!

"Two thieves came walking by that way,
      Rumsty Ho! rumsty Ho!
Two thieves came walking by that way,
And they came to the place where the old man lay,
      Rumsty Ho! rumsty Ho!

"They stole his wallet and they stole his staff,
      Rumsty Ho! rumsty Ho!
They stole his wallet and they stole his staff,
And then broke out in a great horse-laugh,
      Rumsty Ho! rumsty Ho!"