but at least it didn't occur to him that any music theory originating east of Vienna might be relevant to his project.
Did anyone before him who was looking at folk tunes from the British Isles do any better?

Dunno. Sharp got the idea of using the Renaissance modal system from Lucy Broadwood, and I think he was more critical in the way he went at it than she was. Very thorough descriptions of the Arabic modal system were available in French, and d'Erlanger's massive treatise must have been in libraries Sharp used, sitting there too big to ignore.


Do Henry Larcombe's multiple variations (pages 21 and 22 of "Some Conclusions") have any of the character of 'eastern' improvisations within the mode?

I don't have a copy of it handy (I seem not to have put it on my tablet) but I doubt you'd get much of that from an English singer. Construction of fixed melodies using modal principles is another story, and even when there is no conscious awareness of how it's done, people making up tunes follow modal rules.