It's certainly been reported very widely in America, but has been common enough in England (My Boy Billy/Willie or Billy Boy, usually) and in Scotland (My Boy Tammy).

There are a few broadside editions of the Scottish form (which is a re-write made by the poet Hector MacNeill, first published in 1791) at Bodleian Library Broadside Ballads:

My Boy Tammy

There is useful information in Iona and Peter Opie, The Oxford Dictionary of Nursery Rhymes, under no. 44 (Billy My Son/Lord Randal) and no. 45 (My Boy Billy). The Opies don't seem entirely convinced about the Randal connection. Bronson considers it a "spirited parody" of Randal. James Orchard Halliwell found versions in Suffolk and Yorkshire prior to 1844: the Opies quote the Yorkshire text. The Broadside set quoted above reads a little like a parody of the parody, so to speak, though the last verse at any rate has been found in oral currency.