Figuring out the effect of a snake bite gets complicated. Full disclosure: I am a former zoologist.
As an aside, my professor for Herpetology (reptiles and amphibians) had a hilarious story about a guy who needed to answer the call of nature in the woods. He dropped his pants and hung his butt over a log to do so, but the rattlesnake resting there took objection to being dumped on and that butt offered an easy target.
Anyway, the guy lived even though his hunting buddy declined to suck the venom out from the wound.
Some factors involved in what happens after a venomous snake bite:
1. Did the snake actually inject venom? As another poster has suggested, venom is metabolically expensive to produce, and snakes instinctively conserve it. Many warning or defensive bites from poisonous snakes, in other words, not aimed at prey, do not include the injection of venom at all. That's how the guy in my professor's story lived.
2. How deep was the snake's bite? Did both fangs get in or did one or more simply scrape the surface? Also, contrary to widespread belief, not all venomous snakes have hollow fangs. Some get the venom in via flow along a side groove in the fang. As a result, the "quality" of the bite can make a big difference in how much venom makes it into the target.
3. What's the size of the snake, and what's the size of the victim? The first controls the maximum amount of venom that might go in, and the second affects how much of an effect, and how quickly, that venom will have. This makes a child much more vulnerable than an adult.
4. What's the nature of the venom? There are the three basic types mentioned by another poster, but most snakes have a mix of components, for example, the venom might be mostly neurotoxic but also have some haemotoxic components. Also, the mix and nature of the components can vary based on the usual prey of that snake species. For example, the venom from those snakes specialized in eating amphibians won't have the same effect on mammals, such as humans, as the venom of snakes that specialize in eating rodents.
5. The extent of the victim's physical activity after the bite, which influences how far and quickly the venom is distributed, and how soon after the bite there is knowledgeable medical attention.
So, if it's a small snake from a species specialized for "cold-blooded" (reptiles and amphibians) prey with a grooved rather than a hollow fang that did not get a firm hit on the girl's hand, and she gets medical attention right away that includes keeping her calm and inactive, it would be quite possible for her to survive even without anti-venom. There would certainly be tissue damage, probably considerable, and it would take a long time to heal. I would be willing to bet, however, that the majority of venomous snake bites of a human child that include the injection of at least some venom would result in death, so you need to have the facts necessary to credibly get your situation out of the likelier result.
One option you have is to use a poisonous lizard rather than a snake. They have only grooved, not hollow, fangs, and need to get a really good bite and hang on for a while before enough venom gets in so the bite will be fatal.