Okay, so this is what is called a thought exercise. As in, you don't immediately quibble about "There are no Neanderthals left today", you actually do the scenario. The point of this exercise is to determine your ability to give your message to a people completely alien from yourself, and to determine the "real world" truth of your religion. Basically, whether the core teaching is down-to-earth enough.
An island appears one day from the mists of time after being gone for centuries and centuries, directly between Asia and North America. Let's call this island Pangu. The people there, however, are not like us humans, for they are purebred Neanderthals. Genetically, not only can they not speak, but they cannot understand language even sign language. The island is filled with a huge mixture of food, from Asia to Africa to Europe to the Americas, and so on. In fact, these Neanderthals long ago, lived on all continents, but sections of each broke away, trapping them on this island. Neanderthals understand food (including bread, cheese, wine, and even stuff like sausage), they understand instrumental music and dance, they understand pantomime, they understand crude drawings, and they understand skits. They cannot sign language, as said before, but they can understand hugs, thumbs up or down, kneeling, and stuff like that.
The Neanderthals worship a mix of Taoism (which they taught to the Chinese), Shintoism (which the Japanese taught to them shortly before their island disappeared) and sorta nature worship. The thing is, while they completely accept Taoism, the other two are kinda spotty, because they don't understand gods or angels or prophets. And no, this isn't a matter of teaching harder, they believe in things of this world, prophets are sent from God, so if you teach about Jesus, they will only understand that he is a man, and that he died for others.
The Neanderthals have learned modern farming and medicine, as well as how to build skyscrapers (just because), but being good Taoists, refuse to build too many cities on their island. So yeah, they're modern, only you and missionaries like you have concluded they are superstitious heathens (unless you're Shinto or Taoist, then you win this game automatically). So, you have decided to convert them. You have to do so without language, only using symbols and physical action.
I have already tested this for Christianity. It passes, not only on performing a skit about the birth, passion, and resurrection of Jesus, but also many of his parables. They could definitely convert to Christianity, though there would be gaps. Judaism, on the other hand, they probably would pick up the festivals or rituals without understanding anything about God.
Would you be able to teach them enough for them to be considered believers? And what would you teach them if your core beliefs were no good?