So, for the 1900s decade, I've not been able to spot any other recordings.

This ad in a 1905 magazine lists the Minster Singers' recordings of 7 chanties.

http://books.google.com/books?id=wH4cAAAAIAAJ&pg=RA2-PA184&dq=%22minster+singers

On article from 24 Aug, 1907 (Tin he Living Age, vol. 36, no. 3294) on "THE TALKING-MACHINE", mentions "sea-chanties" on cylinder. I believe the genre is merely being used rhetorically to show variety and novelty, and, though speaking generically, the Minster Singers recordings were likely what was in mind.

The record connoisseur has a wide choice in which to specialize. He may go in for specimens of famous bands, such as the Scots Guards, the irish Guards, the Garde Republicaine, &c., or he may prefer English, Scotch, or irish ballads; if instrumental solos, then he has violin, pianoforte, cornet, piccolo, xylophone, banjo, concertina, &c., many of them made by worldfamous artists. Of vocal records there are many fine soprano, contralto, tenor, baritone, and bass records on the market, without counting duets, quartettes, choruses, &c. Then there are national airs or specimens of music peculiar to the different countries; Turkish, Russian, Japanese, Chinese records, and many others, are all obtainable. Descriptive records alone offer a tempting field. If the quaint and exceptional specially appeal, then the collector may go in for pygmy records, sea-chanties, &c., ...

The Minster Singers' recordings were evidently renditions of Bradford & Fagge's collection. I don't recall seeing the influence of those versions on later recordings I've heard. Where these recordings listened to by later performers? I would guess the answer is, pretty much, no. Besides, the next set of performers would be following the scores of someone else's presentations. I wonder if (and I guess there isn't) much or any connection between these recordings of the 1900s and those in the first revival of the 1920s.