Steve Gardham wrote: William must have died shortly after this was published.

Chappell was dead before Wooldridge published (Wikipedia says he died in 1888). I assume he's still listed as the author because he was responsible for the songs which were included, and some of the text was still his. And because it would help sell the book.

Really, someone should have produced a third edition with Chappell's text (corrected where possible) and Wooldridge's correct tunes and additional source information. But there probably wasn't enough demand to make it worth it.