If Jim's song is indeed the one you are after (Roud 357), it appears to have been rewritten several times and appears on stall copies in England, Scotland and Ireland. This 19thc version usually starts 'As I roved out on a fine summer's morning'. and has 7-8 stanzas + chorus. I recently came across a Garland version printed c1800 which is obviously Scottish.
'Rocking the Cradle; or Hushy-Ba' it has 6 stanzas + chorus and starts 'I am an old man of three score and ten'.

A later unrelated broadside piece was printed by Such of London 'Rocking the Cradle Boys; or Pull Away Cheerily' fl 'Pull away cheerily, not slow or wearily' 3 stanzas with chorus.

There are earlier related pieces dating back to the 17thc in Pepys and in Pills.

Oral versions have been collected all over the English-speaking world, but as Jim says, it's not a very common song.

Well-known versions on the folk scene in the 60s, Tom Gilfellon recorded an Irish version and Mick Waterson recorded an English version.