As FumbleFingers indicates in a comment above, "children by different fathers" is considerably more common that "children from different fathers" in both U.S. English and UK English. It is also, by more than half a century, the older wording in Google Books search results.
A Google Books search finds six instances of "children by different fathers" from the period between 1800 and 1856, the earliest of these being this summary from "First Report from His Majesty's Commissioners for inquiring into Conditions of the Poorer Classes in Ireland" (July 8, 1835), reprinted in Parliamentary Papers, House of Commons and Command, volume 43 (1845):
Number of bastards baptised annually, 1832–1834 ; they are never supported by the parish ; the mother rarely desert and never destroy them ; but frequently procure abortions by medicines and over-exertion ; the fathers constantly neglect them ; a feeling of honour frequently induce marriages before birth, but never after ; wages are never awarded before birth ; amounts recoverable in actions for seduction and for maintenance ; no variation in the proportion of illegitimate children has been observable since wages were granted ; few cases of women having children by different fathers ; sum that would be allowed for a second or third child ; no punishment is inflicted on the mothers ; ...
The earliest match for the phrase from a book written and published in the United States is from Thomas Nichols & Mary Nichols, Marriage: Its History, Character, and Results; Its Sanctities, and Its Profanities; Its Science and Its Facts:
Women do not seem to differ from men in their capacity to love ; but as they have less freedom, and more disease, we see less variety of manifestation. The maternal function would seem to confine a woman to one lover at a time in the intimate relation necessary to the production of offspring ; but this is not always the fact, for every medical reader knows of cases in which women have had twin children by different fathers. The power of loving two or more men at a time is a simple question of capacity, and is to be decided, not according to any notion of propriety, but as a matter of fact.
In contrast, the first instances of "having children from different fathers occur in the 1910s, one from England and one from Ireland. From Parliamentary Debates: Hose of Commons (1912) [combined snippets]:
They ["mental defectives"] are paupers in the making from the very cradle, and, when the females get to certain age, they are, I am sorry to say, exploited to a very considerable extent by unscrupulous men to their own undoing and to the great increase of youthful depravity in both sexes. Half the girls in our rescue homes are feebleminded and of 15,000 births that take place annually in our workhouses a very great number come under the same category. The Report teems with examples of this sort of women coming in year after year and producing children from different fathers and from fathers perhaps whose names they do not even know.
From Michael Cronin, The Science of Ethics: Special Ethics (Dublin, 1917):
But the polyandric union not only in no way furthers, but it actually sets up a positive impediment to the birth of children. Under polyandry the birth of children is not, indeed, to be regarded as wholly prevented, since the simultaneous conception of different children from different fathers is physiologically a possible occurrence. But such an occurrence would certainly be perilous to one or both the offspring, and therefore the polyandric union is at least to be regarded as an impediment to propagation, as making difficult the continuation of the race.
The first U.S. match for "children from different fathers" is from Leo Kanner, "Habeas Corpus Releases of Feebleminded Persons and Their Consquences," in American Journal of Psychiatry (March 1938) [combined snippets]:
Rebecca A. was released twice on the petition of her feebleminded, syphilitic, sexually promiscuous mother; she acquired syphilis, attempted suicide, gave birth to seven children from different fathers and married first a syphilitic criminal and then a post-encephalitic cripple.
One possible explanation for the idiomatic preference for "children by different fathers" is that the phrase works with the verb fathered (or sired): "children fathered by different fathers." The same is not true of "children from different fathers": "children fathered from different fathers."