How might neanderthals have communicated without the ability to speak?

First, we don't know for sure they did not have some kind of rudimentary language (they mated with Homo sapiens, we know that!)

but they might have used other forms of communication, such as

pointing, doing  something together (look, i want this tool like this (works on the tool ostensively and hands it to the "learner") - this is also called "shared action" or "joint attention"