As fev's answer suggests, the earliest form of the expression involved hairs past freckles rather than freckles past hairs—although the kindred expression "according to the hairs in my wrist" may be even earlier. Eric Partridge, A Dictionary of Catchphrases (1977/1985) has this entry:
two hairs past a freckle. 'When we {in our schooldays, c. 1958–65} were asked the time was, and we didn't have a watch, we'd look at the back of our wrists as if there was a watch there, and say "two hairs past a freckle"' (Levene, 1978) P[aul] B[eale, commenting in 1985]: the idea, if not the actual phrase goes back a decade or two earlier: our school version, earlier 1940s, was 'according to the hairs on my wrist...'
An instance of this wording appears in an unidentified script from The Goon Show, broadcast at some point between October 1954 and January 1956, reprinted in The Goon Show Scripts (1972):
MORIARTY Don't be a fool — this is no time to take a bath, it's getting late.
SEAGOON Nonsense — plenty of time — according to the hairs on my wrist it's only half past ten.
GRYTPYPE-THYNNE (disbelief) The hairs on you wrist say half past ten?
SEAGOON Yes.
One early instance of "hairs past a freckle" (cited in Greybeard's answer) ups the number of hairs to ten. From the California Folklore Society's Western Folklore, volumes 13–14 (1954[–1955?]):
28d. What time is it?
Ten hairs past a freckle (or mole or some feature).
The oldest recorded instance of the joke that I've been able in the form "freckle[s] past a hair" to find is this brief mention in Southern Folklore Quarterly, volume 26 (1962) [snippet view]:
2.72 Two freckles past a hair.
Another fairly early occurrence is in "What Time Is It?" in Lore and Language, volume 3 (1979) [combined snippets]:
L. C. recalls (July 1979) that in the sixties, at a primary school in Portadown, Co[unty] Armagh, a child who was asked the time and did not possess a watch would scrutinize his or her left wrist for a moment and then answer Two freckles past a hair. Strictly speaking this is less a put-put-off than what the Opies term a "crooked answer".
I couldn't find the 1943 instance from the Canyon [Texas] News cited in fev's answer online.