Perhaps I'm just very bad at googling; though I can find numerous articles describing the discovery of Churchill's writings on the possibility of extraterrestrial life, I can't find the original essays.
The article in Nature mentions some particulars:
I was on a visit to the US National Churchill Museum in Fulton, Missouri, when the director Timothy Riley thrust a typewritten essay by Churchill into my hands. In the 11-page article, 'Are We Alone in the Universe?', he muses presciently about the search for extraterrestrial life.
He penned the first draft, perhaps for London's News of the World Sunday newspaper, in 1939 — when Europe was on the brink of war. He revised it lightly in the late 1950s while staying in the south of France at the villa of his publisher, Emery Reves. For example, he changed the title from 'Are We Alone in Space?' to 'Are We Alone in the Universe?' to reflect changes in scientific understanding and terminology.
According to a later paragraph:
To the best of Riley's knowledge, the essay remained in the Reves's private collection and has never been published or subjected to scientific or academic scrutiny.
I can't locate a version of the original essays online. Are these available somewhere in digital or printed form? Have they subsequently been published since rediscovery in 2017?