The words flour and flower do not only have the same root, but they were also spelled the same until around 1830. (See etymonline.)
Not only did Shakespeare rhyme hour with flower, but he also sometimes spelled them the same.
It fears not policy that Hereticke,
Which works on leases of short numbred howers,
But all alone stands hugely pollitick,
That it nor growes with heat, nor drownes with showres.
The meter requires that most uses of flower I found in his works be pronounced with one syllable.
To thy faire flower ad the rancke smell of weeds,
but there are a few uses that must be pronounced with two syllables:
Flower of this purple die,
Hit with Cupids archery,
Sinke in apple of his eye,
When his loue he doth espie.
Thomas Gray, writing one hundred and fifty years later, does the same thing: hour and lower rhyme:
While Hope prolongs our happiest hour,
Or deepest shades that dimly lower
and flower always seems to be one syllable:
Full many a flower is born to blush unseen,
And waste its sweetness on the desert air.
So it appears that they have been homophones, at least in the London dialect, for over four hundred years.
Wordsworth (1770–1850) also appears to have only used the one-syllable pronunciation of flower, but Tennyson (1809–1892) used both the one- and two-syllable pronunciations:
The purple flower droops; the golden bee
Is lily-cradled: I alone awake.
Both of these poets rhyme flower and hour.
My guess is that if somebody currently uses two syllables for flower and one syllable for flour, the most likely reason is that their pronunciation has been influenced by the spelling.