The answer: an azalea
The pure Korean word for "azalea" is 진달래, and the pure Korean word for "cuckoo" is 소쩍새, but, interestingly, at least to me, the Sino-Korean word for both "azalea" and "cuckoo" is 杜鵑 (두견). Except for context, the only way to distinguish the meanings of the two would be to add the Chinese characters for "flower" and "bird" to the end of the word, such as 杜鵑花 (두견화) for "azalea" and 杜鵑鳥 (두견조) for "cuckoo," though Koreans would say 두견새 instead of 두견조, using the pure Korean word for bird (새) instead of the Sino-Korean (鳥 - 조). Anyway, I wonder how the same two Chinese characters came to be used for both a flower and a bird.
Individually, 杜 (두) is a kind of "pear tree" and 鵑 (견) a kind of "cuckoo," which would suggest to me that when Koreans hear the word 두견, the bird comes to mind before the flower. And that may suggest that Koreans may think of the 杜鵑花 (두견화) as the "cuckoo flower," maybe without even realizing it refers to an azalea.