Most people use a tune from somewhere else, but it's important that they say what they've done. Martin Carthy is an example of the intelligent matching of traditional melodies to texts with which they were not historically associated; he always makes clear his source, but that doesn't prevent people learning songs from his records and subsequently teaching them to others as authentically traditional, which can be irritating. The current short series on BBC Radio 4 about border ballads featured The Twa Corbies last week; sung throughout to the Breton tune first set to it in the 1960s, and with no mention that this was not its traditional tune! The presenter should have known better.

The wording on the website I mentioned is ambiguous and will probably result in people believing that Sharp actually found Robin Hood and Alan a Dale being sung to The Saucy Sailor, which he certainly didn't. Any connection between Robin Hood in the greenwood and Robinhood is to the Greenwood gone can only be conjectural (unless someone who knows more can help), so on balance I'd say, go for the tune Rimbault used, with a note stating the source.

The book was The Robin Hood Garlands and Ballads, with the tale of the Lytell Geste. A collection of all the poems, songs and ballads relating to this celebrated yeoman, by John Mathew Gutch (1847). As I said, I don't know on what grounds E.F. Rimbault, who supplied the musical examples, used it, but with the strong caveat that it's a mid-nineteenth century setting, and likely not traditionally associated with the ballad, it might do for the time being. I've made a midi from the notation given in Chappell's Popular Music of the Olden Time (1859); the first half of Drive the Cold Winter Away, as he stated. Of course, I don't know if this is exactly as Rimbault quoted it, but I'd guess that Chappell's example might be closer than those given in Simpson's The British Broadside Ballad and Its Music (1966), the second of which, though the same melody (from Playford's Dancing Master of 1651), is phrased a little differently.

Here's the midi:  Alan-a-Dale.mid  See what you think. In two places I've split notes into two of equal value in order to accommodate the lyric.