Conducting Factor Analysis on Data with Nominal, Ordinal, and Interval Data

I would advice on conducting a confirmatory factor analysis on data from an instrument that contains nominal, ordinal, and interval data. I'd think to use, e.g., point biserial, polychoric, and product-moment for each respectively, but what about when they're all in the same CFA?

Other posts and articles address factor analysis with ordinal data and nominal data through latent class analysis, but my data contain all three types.

Perhaps finite mixture modelling can do this? Again, I'm not sure, though, how to handle both nominal and ordinal together.

Bonus question!
I know Mplus and LISREL both can handle ordinal data pretty well, but I'm not sure how well these methods also handle nominal, and I would prefer to use R to conduct this if possible; so, I would also appreciate any advice on using R for this if possible.