"Does anyone know if a tune still exists for a version that actually mentions Sir Walter Raleigh? And, if so, where might I find it?"

The only version that mentions Rawleigh (sic) - and then only as the ship-builder, not a protagonist - is Child's A version, a broadside from the Pepys collection of the late 17th century (whoever wrote the Wikipedia entry dates the original broadside around 1635). Apparently there was a tune specified: 'The Sailing of the Low-Lands', but since Bertrand Bronson was unable to find it, I doubt whether you or I would stand much chance.

Incidentally, this early version specifies 'The Neatherlands' as the location of the events, before reverting to 'The Lowlands' in later verses.