It's an invention for the movie:
TheWrap put the question to “Darkest Hour” screenwriter Anthony
McCarten, who not only wrote the screenplay but has published a
companion history book about the events of May, 1940 — and in the
book, he does not describe the scene in the Underground.
And McCarten admitted that no, it probably did not happen. But
something like it might well have. “This is the kind of thing he did
right through the war,” said McCarten of Churchill. “He would go AWOL,
disappear and pop up somewhere in London with ordinary people, to find
out what they were thinking. So that scene was drawn from deep
research, but we have no record that it happened.
“It’s a perfect example of how you’re trying to dramatize verifiable
events that might have happened outside the time frame of your movie,
but which are very, very valuable for the dramatist in showing
critical aspects of your story.”
Slate points out a deeper issue with the scene:
Historian Richard Toye undertook a massive archival dragnet that found
the British did not, in fact, snarl along with Churchill’s speeches.
Upon hearing them, some were inspired, many were dubious, and many
looked to their family and neighbors to assess what they’d just heard.
They didn’t cheer like Minnesota Vikings fans; they, in fact, thought
pretty hard about what the speeches meant. This is another heartening
example from history: Not only did Britain make the hard choice, they
didn’t make it in a fit of rhetoric-induced adrenaline.