Knowing nothing about this topic, I looked it up, and it seems New Orleans was one of the first places in the US where Cuban jazz made an impact. This was in the early twentieth century.
Ragtime music, with syncopated rhythms, existed before jazz and is more sheet music. Ragtime was popular in the late 1800s and early 1900s. Inspirations included marching band music, African beats, European stuff, minstrel music, and others. Kinda like a gumbo of music. So there are many influences, including Spanish.
Composers like Jelly Roll Morton helped the bridge the gap from ragtime to jazz, which is importantly improvisational. Jazz took off in the 1920s in New Orleans. Some of Jelly Roll Morton's music included Spanish influence, and thus similar beats to Cuban music.
So yeah, there are inspirations from Spain, Cuba, the Caribbean, etc on jazz music in New Orleans, but Afro-Cuban did not evolve until like 20 years later in the 1940s. It was brought forward by people like Mario Bauza, born in Cuba who moved to New York, and also Dizzy Gillespie who was important to bridging gap from what evolved as what we might call traditional jazz to Afro-Cuban jazz. And these developments happened in New York, not New Orleans. Afro-Cuban jazz music is ultimately distinct from what one would have heard in New Orleans. Desi Arnaz played a watered down Afro Cuban and if listening to that and the other mentioned musicians I don't think people hear that style of music and think of bayous and New Orleans.
Afro-Cuban evolved partially from New Orleans and other jazz from the 1920s and 1930s, so it did not exist as it's own thing to have been around in New Orleans in the early 20th century. What they share, Cuban influence, heavily influenced Afro-Cuban because Mario Bouza, a Cuban, had a big role, but in what's now New Orleans jazz, Dixieland Jazz, etc, Cuban music was a smaller influence.
I hope that makes some sense. Tried typing out on my phone and I'm feeling lazy to edit.
I'll add that I won't claim to be an expert and that this is all just from my understanding.