Sharp certainly paid William Kimber when he collected the first tunes in 1899. William Kimber also got paid to go to London to teach morris dancing enough to cover his lost wages and expenses. I've read quite a few instances of sources refusing to sing for collectors unless they paid up front (and, as I recall, they weren't cheap).

Sharp wrote piano accompaniaments for the songs to enable them to be taught in schools for which he got paid. I don't know whether he got, or legally needed, the sources' permission to do this; as Richie says the sources didn't write the words or the tune.

To my mind the words and the tune were, and are, public domain and would not be subject to royalties for the source. However, a sound recording of the source's performance is different and should be a source of income for the performer (and pretty much a source of dispute about money between performers, record companies ect ever since).