Jumping the broom, broomstick, besom, turns out to be a tough one to pin down. In Britain, a jump or jumped marriage (hasty or what we would call common law now) goes way back, but the earliest English reference I can find to jumping the broomstick is Macauley 1824; and in American usage 1856; but in both cases the authors use it familiarly and one suspects it is much older.
Since in the States it probably would be the slave owners applying the term, it could well have an English origin.