I haven't read "The Mikado." But the description you have there sounds more like the popularized and mystical "origin of life"/"cosmic evolution" of Robert Chamber's Vestiges of the Natural History of Creation than it does any truly scientific account (e.g., Darwin's work, which doesn't go back as far as that anyway).

Chamber's work basically attempted to tie together a number of notions about the materialist origins of universe and the unity of life in a vague and not especially scientific way. It was however immensely popular and well-known in the 19th century, and there were more copies of Vestiges than Origin of Species floating around until well after Darwin's death. Though largely unknown today except to historians, it was the best-selling book on evolution (both biological and galactic) in 19th-century Britain.

In any case, these kind of vague, popularized notions were quite common by the late-19th century, especially in Britain, and both pre-date and post-date Darwin. They were, as Darwin himself recognized, important to framing the public debate, and preparing the public ground, for Darwin's own ideas, even while Darwin himself felt that these kinds of popularizations were very misleading and non-scientific.

In any even, by 1885, the idea of evolution was well-rooted amongst the reading British public (which was quite large), both through Chambers and through the debates that followed Darwin's works in the 1860s. The exact language appears to be Gilbert and Sullivan's, but these were recognizable, well-worn ideas at this point.

On Vestiges and the popularization of these kinds of ideas, see James A. Secord, Victorian Sensation: The Extraordinary Publication, Reception, and Secret Authorship of Vestiges of the Natural History of Creation (University of Chicago Press, 2000).

As a little follow-up, was there a general trend in aristocratic individuals not only tracing their family trees but also publicly boasting about this?