Palmer and Spencer influences did not seem to travel well from what I have read, and Vere Foster was covering the need for a simpler style in Great Britain. To be precise Chrurchill's parent were both English (although his mother was American-born English) and his education was English and he certainly was not the intended subject of the song, which looked at an earlier period for comedy.
Sorry, not quite right. Jeanette Jerome was born and raised in New York City and a socialite. Her education would have included writing in what was almost surely one of the American styles. Considering her place in society, 'Ladies' Hand' (a form of Spencerian) is a likely candidate. Lady Churchill was English only by marriage to a slightly pottier than usual British lord. My father was English and one great-grandfather was French. The handwriting of neither is foreign to me. (I will admit I had trouble with a note the Earl of Darmouth once penned me, but his handwriting would probably defeated anyone other than his secretary.)
While it's doubtful W. Churchill wrote in an American hand, Spencer or Palmer or whatever hand Lady Churchill wrote would hardly have been foreign to him, i.e., unknown, strange, or undecipherable. As for Gilbert and Sullivan, I have an advanced degree in vocal performance and am quite aware when they were active. I might point out, however, that Gilbert, like many talented librettists and playwrights, satirized contemporary events and personages by recasting them in earlier settings. His humor was nothing if it wasn't topical. (I'm not a fan.)
That said, I didn't suggest the Winston Churchill was the target or subject of the lyric, merely that a Ruler of the Queen's navy might not necessarily find Spencerian (or any other hand) necessarily foreign.