I think the song must have become part of a medley c1900 when such things were popular. I have collected versions attached to other songs of the same early 19thc era. My own family lullaby (at least 4 generations) has your 2 starting verses, a spoken link 'and this is the story he told' then the tune changes to Bayley's 'No place like home' with 2 more verses of the lad's life story.

I also recorded another version that starts off with our 2 and then launches into 'Poor Little Joe', another similar song of the wandering beggar boy type, typical Victorian tear-jerkers.

There were hundreds of these bloody things printed on broadsides and 'Early Pearly' as we call it derives from a broadside which has a wide range of titles 'Lady and the Sailor Boy' being one of the most common though he is just as likely to be 'The Soldier's Boy' and the adjective 'Poor' is often somewhere in the title. They all start with 'The snow fastly falling or descending'. Both Pitts and Catnach printed versions so it's at least pre 1832, and all the major printers had versions. It is rather similar to the female equivalent ' The Poor Little Beggar Girl' which has just as many variants and was popular in America.

Most variannts have 4 or 5 double verses. There are versions of both on the American Memory website, but for a quick find try the Bodleian Ballads website Harding B11 (3527)The Soldier's Boy. I'm sure you'll find lots of other versions on there.