This isn't really a research question, but at least it's research-level mathematics. I'm talking with some other people about the first uncountable ordinal, and I want some facts to inform this discussion. Specifically, what useful or interesting foundations of mathematics do or don't allow one to prove the existence of an uncountable ordinal?

If you don't have a better interpretation, then for "useful", you can probably take "capable of encoding most if not all rigorous applied mathematics"; for "interesting", you can probably take "popular for study by researchers in foundations". For "existence of an uncountable ordinal", you could take "existence of a well-ordered uncountable set", "existence of a set whose elements are precisely the countable ordinals", etc.

Hopefully there is a body of known results or obvious corollaries of such, since it could be a matter of some work to apply this question to foundational system X, and I don't expect anybody to do that.