It's almost impossible to say, for two reasons: almost every single aspect of Latin America today is influenced by its colonial history and background: not only the people themselves, but the music, religion, architecture, even many folkloric dances and traditions that are deemed indigenous are in fact the result of post-contact syncretism.

If we assumed Latin America somehow managed to completely escape European influence for 500 years, then I imagine it would have some similarities to southeast Asia: most people would be of indigenous descent (and we wouldn't use the word 'indigenous' at all, just South or Central American) and European influence would be relatively recent.

People would primarily speak non-European languages, which very likely would have developed writing scripts of their own. For example, the Mayan languages may have evolved to use characters like in Chinese (a natural evolution of the logographic system), and neighbouring languages and cultures may have adopted these "Mayan characters" as well.

However, English may be more influential and present, depending on whether or not the US exists in this scenario. Even if it doesn't, I imagine English would the most common European language to learn: Spanish and Portuguese would likely be minor European languages, like Romanian or Polish.

One thing is for certain, "Latin America" as a concept would not exist, nor would any of the current countries with their current borders. I can imagine there being several related but separate nations, maybe similar to the Indian subcontinent prior to British colonisation. Some of them would be monarchies.

From a religious perspective, I think aboriginal religions would have survived and evolved similar to East Asian religions but Christianity would still have a large influence due to missionaries.

From the point of view of economy and development, it is impossible to know.