Even if nineteenth-century children didn't quite know how to make exotic animal noises, the evidence suggests they would be making them up.
Although elephants in particular, and more dramatically lions, had been part of the American circus/travelling menagerie scene from its beginnings in the first years of the 19th century, the American West really wasn't. Of course, at that time "West" meant today's Midwest, and perhaps only the eastern parts of it. But even in 1866, Dan Rice's Circus was complaining (or bragging?) that it cost them $50,000 to go all the way to Wisconsin. Exaggeration--perhaps; that is the bread and butter of circuses. But Rice is later noted for having lost $15,000 on a single season's tour, so one should attribute at least a sizeable sum to the expense.
But children who never saw a circus with an exotic menagerie (horse-and-pony trick shows were extremely popular), who never travelled east to the Lincoln Park Zoo (Chicago) after its opening in 1868, almost certainly took an interest in making animal noises.
What's often called the earliest picture book, the Orbis sensualium pictus, features an at-the-time standard component of instructional books for children--an A-B-C. The Orbis pictus is actually a language instruction book, designed to teach Latin. It was originally written by a Czech author in the seventeenth century, with English (and other, including polyglot!) translations following. Its A-B-C matches each letter of the alphabet with an animal, providing the vernacular and Latin names. (The link between animal and letter is a sound that occurs somewhere in the word, not necessarily the first letter).
The Orbis pictus is crucial here because it doesn't just list animals. It classifies the animals by the sounds they make:
Cornix cornicatur: the crow crieth
Agnus balat: the lamb bleateth
Cicada stridet: the grasshopper chirpeth
and so forth. One English translation of the Orbis pictus was indeed printed in the U.S. (New York City, 1810), but I'm not really concerned about the chances of any one child getting ahold of this one edition of one book. Instead, I--and more to the point, other scholars--would point to the influence of the Orbis pictus on subsequent children's literature. The animal ABC/animal sounds tie to nursery-room books remains strong today, seeming "only natural." Well, it's not natural: it's Johannes Comenius' humanist legacy at work!
By the later nineteenth century, we can see the phenomenon of animals and animals noises all over the place in the exploding genre of "children's literature." German author/publisher Lothar Meggendorfer's "action books" (featuring pop-ups, tabs to pull, threads to lift, etc) made great use of inviting readers to fill in the noises made by animals splashed across their pages, including fearsome lions. Popular nursey rhymes like The Three Little Kittens used animal noises like kitties' plaintive meowing as a refrain of sorts.
Margaret Gatty's popular Parables from Nature is filled to the brim with stories starring animals. I want to focus on the vignette called "Inferior Animals," though. It focuses on one animal specifically, the rook (a bird) and does mention cawing as an onomatopoeic sound. More interesting for present purposes, however, is the overall theme of the parable. It's not "what does the rook say" in the sense of cawing, but what does the rook say in the sense of what does it mean by the cawing. In a sense, Gatty links the real world of actual animal noises with the common children's literature trope of anthropomorphized talking animals.
But why I bring up Gatty's "Inferior Animals" actually has to do with one final, crucial step: expanding beyond the pages of a few rareified children's books to the population of children more broadly. Gatty's concern to guide children towards "interpreting" the sounds of animals finds deep resonance in--of all places--late 19th and early 20th century children's psychology research. Really! Psychologists, while on one hand trying to gain credibility for their subject as a whole, were investigating child development quite eagerly. They were utterly fascinated by children's tendency to interpret the "language" of cats and the body language of dogs. (It was viewed as part of the arrival at a concept of "self" and "human" distinct from animal).
In other words, by 1900, American children were interested enough in animal noises to draw the attention of researchers who really needed every study to pay off so their field would look better. This suggests that even if their parents had shoved them in front of a New England Primer instead of an Orbis sensualium pictus ("The idle Fool / is whipt at school; As runs the Glass / man's life doth pass; ...Job feels the rod / and blesses God; proud Korah's troops / were swallowed up"), they were experimenting with animal sounds at play if not in lesson.