I'm going to try to make a few notes on some of the chanteys that have been mentioned.

I don't think many of the Caribbean chanteys that appear in Abrahams' book (to the extent that he also made recordings of them), or that we find reintroduced by the Nevisian whalers or in Lomax's recordings represent a broken link in oral tradition -- precisely because we have these as aural sources directly from an oral source. Many of them are indeed less known (though note the fact that, for example, that Lomax's recordings in the Bahamas, mentioned by Fred above, were drawn from by early folk revival groups). While they are less known, we have a chance of knowing them directly from an oral source.

That not only Hugill, who by happenstance got many chanteys from Black chanteymen in the Caribbean, but also Sharp, Terry, etc. collected some of the same chanteys (by name, at least) from Englishmen suggests that these were a stock of chanteys that were widely known among the international set of sailors. This may be a bit different than localized chanteys like "Feeney Brown" and such. It's the former set that I think is "extinct" or "endangered," despite us having them in print.

Then there is the category of really "lost" chanteys that we don't have much knowledge about because they have only been named in books, without many lyrics and without melody. greg stephens brought up the example of one of these that seems to have been rediscovered. Here is his thread on that topic. "Fire Maringo" would be another that could fit into that category. The issue with these is that, unless and until they are rediscovered, they remain as vague references. By contrast, the many less-known chanteys in Hugill et.al. are at least notated out.

So adding some of the chanteys that have been mentioned…

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"Coal Black Rose"
Print: Hugill
Performers: Tom Sullivan (recorded on "Salt Atlantic Chanteys); Finn & Haddie (here)

Notes:
Learned by Stan Hugill from Harding. It is another one of those chanteys which can be presumed to be developed from a snatch of a popular minstrel song. In this case, the song "Coal Black Rose" is believed to be one of the earliest minstrel songs, popularized by George Washington Dixon by 1829 and henceforth part of the standard repertoire of blackface performers. A copy of the sheet music to the song can be seen here

The opening phrases, especially about the banjo, with the onomatopoeia (as well as some of the melodic contour, I think) are similar enough to make it probable the chantey is partly derived from the popular song.

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"Come Down, You Bunch of Roses"

Print: Hugill, in several books; Doerflinger
Performers: hmm…very many, but few. See below

Notes:
Despite it being apparently very well known, I think this one belongs to this category of broken-oral-link chanteys. Doerflinger's printed version comes from another text. Hugill's come from the oral source, Harding, but so far as I know he hasn't passed that version on orally to others. In fact, he appears to later forsake Harding's version (in SEVEN SEAS) in favor of a popular revival version.

I've tried to analyze what I think is the trajectory of this chantey, since the end of sail, here. The gist of my argument is that Bert Lloyd refashioned the song, from who knows what, and that his version became so ubiquitous that even Hugill adopted it.

Now, we do have those Caribbean recordings of a sibling or parent version of the song, as noted by Fred and Barry. So there is an oral source. One question would be whether these are the same as the developed chantey.

I've made an attempt to recreate Hugill's original/Harding's version here

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"Bully in the Alley"

Note that this is different than the "Shinbone Al" mentioned earlier in the thread.

At first glance, one might not include this one among the "broken link" chanteys of this thread, simply because Hugill recorded it. It is available to many, and many have learned it. In fact, it is so well known (in a form) that it has reached the level of Disney-fication, here and here

However, two points. One is that Hugill's notation of this chantey in SEVEN SEAS is significantly different from the popular versions. I don't have his recording, so I can't tell if the popular versions are based on that (which differs from text) or if they derive from someone's reworking of it. Second, see TomB's points, above, that John Short's rendition had some significant differences. So there is a case for saying that the oral tradition of this chantey has been broken, or at least that the way most people know it now is "off." Who popularized the current version going around (was it Starboard List, as Lighter mentions)?

Regardless, it is similar to a lot of these other chanteys.

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"Lowlands Low"

Print: Hugill
Performers: Ian Cambell; Mystic Seaport Chanteymen; Roll & Go

Revived in the 60s by Ian Campbell and Co.? This is post-SEVEN SEAS, but I don't know if he learned it from print or oral.
Incidentally, there are a number of recordings, by revival singers in the 60s-70s, that are recreated from text. Often they have "mistakes" in them, imperfect readings of the texts, and through their spread on record they've spread these off versions. An example might be Louis Killen's version of "Hilo Johnny Brown" which seems it must derive from Hugill but has too many inaccuracies. It may well be that he learned it from Hugill directly, in which case – sadly all too often the case—it's Hugill's notation that is way off.