There is a persistent confusion between two different poppies with two different histories, and it is worth separating.
Papaver somniferum, the opium poppy, is the plant of the alkaloids, generally with large glaucous leaves and pale lilac or white flowers. Papaver rhoeas, the corn poppy or field poppy, is the scarlet weed of disturbed arable ground. It is not a source of opium and contains only trace amounts of unrelated alkaloids.
The remembrance poppy is rhoeas. Its association with the dead of the First World War is specific and documented: the species germinates prolifically on freshly disturbed soil, and the churned ground of the Western Front produced extraordinary flushes of it. John McCrae's poem In Flanders Fields, written in 1915, fixed the image; Moina Michael in the United States and Anna Guerin in France turned it into a wearable emblem and a fundraising device from 1918 onward, and the British Legion adopted it in 1921. That is a chain of named individuals across six years, not a folk tradition.
The older symbolism of the poppy as sleep and death is classical and attaches to both species without much distinction. Poppies appear with Demeter in Greek cult, they grow in grainfields, and the goddess of grain acquires the weed of the grain, and with Hypnos and Thanatos, sleep and death, who are brothers in Hesiod. Both readings, the agricultural and the funerary, are already present in antiquity, and the modern remembrance poppy sits at the junction of them by accident rather than design.