"Fell" is an adjective and not related to the verbs "fell" or "fall"
It means "strong and cruel". It is rare in this sense, except in the expression "one fell swoop". Tolkien uses it to describe some of his monsters:
The fell beasts were winged creatures with beak and claws, similar to birds but much larger than any other flying beast.
The phrase "one fell swoop" is from Shakespeare. Macduff refers to how a red kite swooped down and killed all the chickens and chicks, as a metaphor for how Macbeth massacred his family.
Did you say all? O hell-kite! All?
What! all my pretty chickens and their dam
At one fell swoop?
-- Macbeth, act IV
The sense of "cruelty" in the original phrase has been lost, and most native speakers won't know the word "fell" in this context. For most native speakers of English it just means "suddenly and in one attempt".
See this proverb described at phrases.org.uk