Unlikely, a very strange happening, but not impossible.
First thing, let's determine how old your baby is, and the differences between an elephant baby and a human.
Baby elephants walk pretty quickly, and human babies don't. It can take anytime between 6 months and two years for a baby to walk (with the average being between 9 months and 1 year). What this means is that your human baby doesn't have the stamina to keep up, and herds move quite a bit. They won't likely have it even when they can walk, so baby would have to be carried a lot--and that would not be hard because elephants can lift a lot with the trunk...
Most cross-species adoptions involve animals with more base similarities, than is true here, especially when it comes to how their young develop--and baby elephants aren't nearly as helpless as humans are by any measure, for any real amount of time. At least with wolves or lions, the pups/kittens are helpless for more than a couple of hours, so they have the instinct to deal with it--if not the milk supply.
And the biggest hurdle is size, which is why I think a toddler would be better than a newborn or a baby, preferably a TALL toddler. I'd say you might want to age your kid up--4 years old, 5 years old, at least.
A newborn elephant is 3 ft tall at birth and while a mother might try to help her baby stand and nurse, they won't necessarily know how to deal with a 6 month old human baby, which will be considerably shorter and who can't walk with the herd.
In Africa, elephants can cover over 50 miles (80 kilometers) in a day, if food is scarce, but rarely walk that far. More often they cover a few miles during the day, and sometimes spend most of their time near a water source. The distance that elephant herds travel seems to change during the year.
That is something you want to consider.
Now there's this lion thing you've thrown in--I actually find that a trifle more likely, TO BEGIN WITH, on the kid being baby age. Lionesses have been known to protect babies not of their species just by instinct, from male lions--she'd have to be tough, wary, and ever-watchful, and it would help if the human had some sibs growing up with her/him. The vitamin C deficiency may never be a problem, actually. There's some argument that that raw meat actually contains vitamin C and some of the organs in the body are surprisingly a good source of it.
The true hurdle is how long a baby is helpless, compared to animals, both with the lions and the elephants. It takes us forever to even walk. The upside to your elephants is the long weaning period (up to 10 years sometimes), whereas your lions don't nurse that long--and they start teaching cubs to hunt at 3 months--which, good luck to the baby...and they wean between 6 and 10 months. Humans do the same but...we don't have teeth for a while, and even once we do, we don't have very many for a long time and they aren't up to the task of raw meat. Neither is the digestive system, so again, I'd age up your kid to like 5 years old or something instead of baby aged.
Let's go through the point-by-point on your elephant advantages:
- Protection from predators, most lion prides won't even bother with
elephants Cross nursing, one of the elephant documentaries I watched
was about an orphaned baby elephant being raised by a herd other than
the one that had its mom. Weak baby elephants still sometimes get picked off by predators who bide their time.
- Prehensile trunks The advantage here is that your elephants could carry baby without harming them, whereas, a lioness would try to carry by mouth.
- Rich milk Good point, but your baby has to be tall enough to reach it. Maybe mom can use her trunk to shove baby up there so it can reach? That'll be awkward but not totally impossible, I guess?
- Very little risk of illness(at least compared to being raised by
carnivores)I respectfully disagree. There will be plants that baby
or toddler is not equipped to digest or might be poison to humans but
fine for elephants. This is a human in the elements, the risk of illness from something eaten is higher than you might think.
- Injury risk is low Now, you talk about clawing wounds from lions--but--elephants play as well (and stampede--no idea what a crawling baby could do in a dust storm of stampeding elephants, who have the instincts not to run over their several hundred pound babies). Cats have signals used when play gets too rough, as do elephants, but a baby elephant is actually more likely to accidently kill a tiny human than a lion kit would, simply because of the difference in size. Adult elephants can be very gentle, but it is something that they have to learn. Imagine rough and tumble play with a group of 3-4 or 200-lbs babies and who are playing together and a 30lb child comes in between it. Accidents, broken bones, maybe even death can result.
- Can sense vibrations through the ground as well as hear I am not certain how this would benefit the baby
You list your pros, but NO cons, so--
- getting the runs and dying from it from bad hygiene, eating something your species can't tolerate
- parasites (just by dint of being in the wild)
- 200 lbs playmates who might accidentally squash you like an insect when you are barely able to walk.