It is consistent with $\mathrm{ZF}$ that a countable union of countable sets may be uncountable. As far as I understand it, this is because in absence of $\mathrm{AC}$ we cannot necessarily choose a bijection between each set in our family and $\omega$. (As this entails infinitely many choices.) I imagine the situation might be somewhat different with countable sets of sufficiently small countable ordinals, since for these such bijections may be described by some kind of formula.
Since there are still many pretty big creatures among countable-in-$\mathrm{ZFC}$ ordinals (while casually reading about ordinals on the internet, $\omega_1^{CK}$ in particular caught my attention, because of its definition in terms of computability) I am wondering:
What is the smallest possible ordinal $\alpha$, for which it is consistent with $\mathrm{ZF}$ that $\alpha$ is uncountable?
In other words, what is the smallest possible ordinal $\alpha$ (describable (I guess) in some elementary terms other than "$\alpha$ is the smallest uncountable ordinal" and possibly countable in $\mathrm{ZFC}$) such that it is consistent with $\mathrm{ZF}$ that $\alpha=\omega_1$. I hope this is not too vague ...
For example: is it consistent with $\mathrm{ZF}$ that $\varepsilon_0=\omega_1$ or perhaps $\omega_1^{CK}=\omega_1$? What is the best result known along these lines?
Just curious. Thanks.