I've probably looked at as many period versions of "Drunken Sailor" as anybody, and I don't recall any of them indicating the "eye" thing.

I think Hugill was full of it. His claim that it was "always" pronounced that way is demonstrably wrong. We have access to all the text sources he had, and more. We don't have access to his lived experiences, but there is no compelling reason IMO to think he had any extensive exposure to this song in its original tradition (considering how much he relied on published texts). Even if he had heard some "authentic" singer do that, a single anecdote wouldn't mean much, especially without citation.

On the other hand, "Drunken Sailor" had already been "revived" and sung by "gentlemen" in their clubs before Hugill even went to sea; it was better known in Hugill's time as a popular song than a laborer's custom.

Burl Ives, on the other other hand, published it as "earl-eye" in 1956.

Sounds like folkie folklore to me.

Stuart Frank repeats the "it was always pronounced earl-eye" folklore in a 2000 publication. When I read that I felt like shaving his belly with a rusty razor.